Mechatronics 15 (2005) 703–729 Control architecture for the pneumatically actuated dynamic walking biped ‘‘Lucy’’ Björn Verrelst *, Bram Vanderborght 1, Jimmy Vermeulen, Ronald Van Ham, Joris Naudet, Dirk Lefeber Vrije Universiteit Brussel, Department of Mechanical Engineering, Multibody Mechanics Research Group, Pleinlaan 2, 1050 Brussels, Belgium Received 20 July 2004; accepted 20 January 2005 Abstract This paper reports on the control structure of the biped Lucy. The robot is actuated with pleated pneumatic artificial muscles which have interesting characteristics that can be exploited for legged machines. They have a high power to weight ratio, an adaptable compliance and they reduce impact effects. The considered control structure can be divided into two parts: a real-time joint trajectory generator and a trajectory tracking controller. The latter consists of a computed torque controller, a delta-p unit, a PI controller and a bang–bang pressure controller. The trajectory generator provides polynomial joint trajectories while the computed torque in combination with the delta-p unit calculates the required muscle pressure levels. The PI and bang-bang controller work at pressure level to cope with modelling errors and to set the pressures in each muscle. The proposed control architecture is evaluated with a full hybrid dynamic simulation model of the biped. This simulator combines the dynamical behaviour of the robot with the thermodynamical effects that take place in the muscle-valves system. The observed hardware limitations of the real robot and expected model errors are taken into account in order to give * Corresponding author. Tel.: +32 26292863; fax: +32 26292865. E-mail address: bjorn.verrelst@vub.ac.be (B. Verrelst). URL: http://lucy.vub.ac.be (B. Verrelst). 1 Supported by the Fund for Scientific Research Flanders (Belgium). 0957-4158/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechatronics.2005.01.002 704 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 a realistic qualitative evaluation of the control performance and to test its robustness. At this stage a controller is designed for and evaluated during the single support phase only. In the double support phase the robot is over-actuated and consequently other control strategies are necessary. Additionally, preliminary results of a walking motion of the real robot with both feet in the air are given. These show promising results concerning tracking performance of the proposed control architecture. The experimental results confirm that the pneumatic tracking system can be used for a dynamic application such as a biped walking robot. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Legged robots; Pneumatic artificial muscle; Dynamic control; Hybrid simulation 1. Introduction 1.1. Legged locomotion and pneumatics The last two decades, substantial progress has been made in the field of legged robots. One of the important and astonishing events happened in Tokyo on December 20th in 1996 when Honda Motor Corporation presented there first humanoid ‘‘Honda Human Robot’’ [1] after 10 years of secret research. This autonomous humanoid is able to walk smoothly on level ground and even to climb stairs. This event seems to have triggered a robot technology race between the big Japanese companies to develop their own legged walking machine. Honda has already built several humanoids of which the latest model is ASIMO [2]. Sony brought to the marked the dog petrobot AIBO [3] and recently revealed the ‘‘First Running Robot: QRIO’’ which belongs to the family of small humanoid robots SDR4X-Sony Dream Robot [4]. Toyota Motor Corporation announced its ‘‘Partner Robot’’ [5]. Fujitsu introduces a miniature humanoid robot HOAP-1 [6] and Kawada Industries developed the state of the art HRP-2 [7] humanoid platform. The latter has been built within the framework of the HRP-project [8] of the Japanese government. This project tries to create ready-to-use industrial, domotic and health-care applications for humanoid robots. This evolution brings the idea of humanoid robots out of science fiction into the real world and stimulates other research institutes all over the world to invest in this highly multi-disciplinary research area. The number of robots built by universities and governmental technology centres is increasing rapidly. Each of these models focus on their own specific implementations going from biological mimic for control purposes to autonomous navigation by means of stereo vision. Despite the wide variety of the projects there is one thing that almost all existing robots have in common: electrical actuation. Motor drives and their attributes are after all widely available and the control aspects of such actuation is well known. But there are some important limitations to this kind of actuation used for legged machines. Since rotational speeds of the inter limbs joints are much lower than nominal speeds of most electrical motors, transmission units are almost always required. These increase the weight and complexity of the actuation system and induce high reflected inertia. The latter is B. Verrelst et al. / Mechatronics 15 (2005) 703–729 705 awkward for shock absorbance, especially when robots are expected to improve in velocity and mobility. For manipulator robot implementation, stiff joints have always been favorable above compliant joints since they increase tracking precision. For legged robots however, tracking precision is not that stringent as overall dynamic stability. Elastic joint properties can then be exploited to store motion energy, reduce control effort and energy consumption. In this context pneumatic systems are an interesting alternative for the actuation of legged robots. Mainly two types of pneumatic actuation are used: pneumatic cylinders and the biologically inspired pneumatic artificial muscles. Standard pneumatic cylinders are well known and available off-the-shelf, but pneumatic artificial muscles have the advantage not having to tackle with the inherent stick-slip phenomenon and are controlled by pressure levels instead of air flows to influence position. Both systems have a high power to weight ratio and can be directly coupled without complex gearing mechanisms. Due to the compressibility of air, a pneumatic actuation system inherently possesses a compliant behaviour. This reduces shock effects at touch-down of a leg. In some pneumatic configurations, joint compliance can be adjusted by applying appropriate pressure combinations. This compliance adjustment influences energy consumption and control effort by adapting the natural dynamics of the system as a function of the desired robot motion. In spite of these benefits, pneumatic actuation has not been widely used as actuation system for legged locomotion, mainly because of its extra design effort needed to control these systems when used for a dynamic control application. 1.2. Pneumatic legged robots: Overview The number of pneumatic legged robots built world wide until now is rather limited compared to the amount of robots with electrical actuation. One of the first to incorporate pneumatics is the Japanese pioneer for legged locomotion Kato. During the 1960s and the 1970s he has built several statically balanced walking bipeds such as WAP I, WAP II and WAP III [9]. These machines where actuated by different types of pneumatic artificial muscles and were able to move very slowly. Another pioneer in legged robotics is Raibert, who has built several hopping and running machines during the 1980s at MIT. His mono-, bi- and quadruped robots used pneumatic cylinders to actuate the telescopic legs [10]. Raibert implemented decoupled control for body pitch and forward speed. The latter was controlled with an intuitive law for foot placement during touch-down. The pneumatic cylinders were used to provide trust action. At the University of Paris, two pneumatic robots were developed: a quadruped robot ‘‘Ralphy’’ [11] and the hybrid wheeled-leg robot ‘‘Sapphyr’’ [12]. Also in Paris the pneumatic biped ‘‘Bipman’’ [13] has been developed at INSERM, and now resides at the University of Toulon. The main objective of this project is to develop a hierarchic and modular architecture able to deal with the different stage of control. Similar to ‘‘Ralphy’’ and ‘‘Sapphyr’’, this architecture is based on a biomechanical analysis of humans at high level and at low level is the trajectory control achieved with computed torque techniques in combination with an estimated force model of the pneumatic actuators. Additionally, a force control on 706 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 the legs has been provided parallel to the joint position controllers. At the Laboratoire de Robotique de Versailles, the pneumatic biped ‘‘STEP’’ [14] was built with pneumatic cylinders. This robot is underactuated and its main focus is the implementation of a new sliding mode control scheme. In Great Britain the spider-like pneumatic robot ROBUG IV [15] was created at the University of Portsmouth and focused on modularity. Caldwell, at the University of Salford, developed the biped ‘‘Salford Lady’’ [16] and a gorilla like robot [17], both actuated with McKibben artificial muscles. The local joint control directly calculates desired pressure levels with a PID position feedback loop, the pressure itself is regulated with fast switching pulsewidth modulated on/off valves. At FZI Karlsruhe, Germany, in collaboration with Karsten Berns, a stick-insect-like robot ‘‘Airbug’’ has been built and its successor ‘‘AirInsect’’ [18] is developed. As the former two, the mamal-like robot ‘‘Panter’’ [19] at FZI is actuated with Fluidic Muscles, which is a type of pneumatic artificial muscle commercialized by Festo. The local control of the antagonistic muscle setup is a series of different controllers. A pure PID position feedback control dictates the necessary contractions, which result in two desired pressures based on a muscle force feedback. The forces are not directly measured, but estimated with a linearized model of the static force to contraction muscle characteristic in combination with pressure measurements. The desired pressures of both muscles are then set by a PI controlled pressure feedback loop. Additionally, the stiffness of a joint can be adapted. Festo itself developed a full scale humanoid ‘‘TronX’’ which is actuated with more than 200 pneumatic cylinders. Although this robot has two actuated legs, it is not able to walk autonomously. Also in Germany, at the university of Lübeck, different pneumatic robots with standard cylinders as FRED II [20] have been built and studied. The robotÕs motion is achieved by orthogonally moving frames. The low level position control of the pneumatic cylinders is achieved with a specific neurofuzzy method, NetFAN approach, in order to cope with the complicated control characteristics of pneumatic cylinders. In the 1990s three generations of bipeds [21] driven by pneumatic cylinders have been built at the polytechnic institute of Czestochowa in Poland. These models were statically balanced and the research focused on simplicity and incorporation of higher level pneumatic control hardware. At the University of Lódź in Poland the small, simple and extremely low cost pneumatically driven quadruped ‘‘Spike’’ [22] has been developed. This robot is actuated with only two pneumatic cylinders and tries to mimic turtle movement. The pneumatics are driven with a simple open loop structure, using pulse-width modulated valve regulation. In Italy at the University of Cassino the pneumatic biped EPWAR3 [23] is studied, this is already the third version of a biped actuated with pneumatic cylinders. Special to this biped is the discrete number of postural positions since the cylinders are used in a binary way. Dynamic postural stability is simplified by using suction caps which fix the feet to the ground each step. Recently in Catania an articulated pneumatic robot leg [24] with two pneumatic cylinders actuating the ankle joint and one cylinder to rotate the knee joint has been developed. A fuzzy controller, which mimics a PID position feedback control, is implemented to regulate the position of the pneumatic cylinders. In the Netherlands at the University of Delft several biped models have been developed by Van der Linde [25] and Wisse B. Verrelst et al. / Mechatronics 15 (2005) 703–729 707 [26]. These robots try to incorporate passive behaviour by exploiting the compliance of the pneumatic artificial muscle actuation system. Active control is added by using so called, phasic activation. At certain phases during a walking cycle pneumatic energy is added to the system by temporarily increasing pressure levels. In the United States at Case Western Reserve University R. Quinn and his team have built several pneumatically powered robots. The first is ‘‘Robot III’’ [27] which is actuated by pneumatic cylinders. For its successors ‘‘Robot IV’’ [28] and ‘‘Ajax’’ [29] the standard cylinders are replaced by pneumatic artificial muscles. All three robot designs are inspired by a cockroach. The swing leg control consists of three different stages. The lowest trajectory tracking is achieved with a local proportional feedback controller and the inverse leg kinematics is implemented with a neural network that coordinate the different joints in a leg. Finally a distributed network is used to control the different legs, which results in a tripod gait similar to the real cockroach animal. The same insect inspired also researchers at the University of Illinois to develop the hexapod robot ‘‘Mark I’’ [30]. This robot is actuated with pneumatic cylinders which are controlled by a kind of pulse-width modulation in order to mimic several features of nerve impulse control for insect muscles. Overall coordination of the legs is done by PID trajectory tracking for which the desired trajectories are constructed with captured data of real cockroaches. A tiny hexapod robot ‘‘Sprawlita’’ [31] has been designed at Stanford University trying to mimic several features of the same cockroach. This robot uses small pneumatic cylinders for primary thrust action. The cylinders are attached to the hip by a passive element, while small motors rotate the legs to direct the thrust action. The control counts on the tuned passive self-stabilizing visco-elastic mechanical system of the legs, while gait characteristics such as forward speed are controlled in a feedforward manner by foot hold placement at touch-down. Recently a pneumatic jumping quadruped [32] has been developed at the Hirose–Yoneda lab in Tokyo. This robot is actuated by pneumatic cylinders and is able to make large jumps. The purpose of this robot lies in the field of rescue operations in disaster areas. And its control currently focusses on the exhaust flows in order to influence impact behaviour. 2. The biped Lucy The Multibody Mechanics Research Group of the Vrije Universiteit Brussel has built a planar walking biped ‘‘Lucy’’. This biped is actuated by pleated pneumatic artificial muscles (PPAM) which are a type of artificial muscle developed in our laboratory. A pneumatic artificial muscle is essentially a membrane that expands radially and contracts axially when inflated, while generating high pulling forces along the longitudinal axis. Different designs have been developed. The best known is the so called McKibben muscle [33]. This muscle contains a rubber tube which expands when inflated, while a surrounding netting transfers tension. Hysteresis, due to dry friction between the netting and the rubber tube, makes control of such a device rather complicated. Typical of this type of muscle is a threshold level of pressure before any action can take place. The main goal the PPAM design [34,35] was to 708 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 avoid friction, thus making control easier while avoiding the threshold. This is achieved by arranging the membrane into radially laid out folds that can unfurl free of radial stress when inflated. Tension is transferred by stiff longitudinal fibres that are positioned at the bottom of each crease. A photograph of the inflated state of the Pleated Pneumatic Artificial Muscle is given in Fig. 1. If we omit the influence of elasticity of the high tensile strength material used for the fibres, the characteristic for the generated force is given by:
 l 2 F ¼ pl f
; ð1Þ R where p is the applied gauge pressure, l is the muscleÕs full length, R its unloaded radius and
is the contraction. The dimensionless function f depends only on contraction and geometry, expressed by the broadness R/l. The muscles used for ‘‘Lucy’’ are all identical and have an initial muscle length l = 0.110 m and unloaded radius R = 0.016 m. The working pressure applied in the muscles of ‘‘Lucy’’ is bounded to about 3.5 bar. Fig. 2 shows the generated force to contraction of the specific muscle for three different gauge pressures: 1, 2 and 3 bar. At low contraction, forces are extremely high causing excessive material loading, and the generated forces drop too low for large contraction. Thus during operation, contraction is bounded between two limits, 5% and 35%, in practise, resulting in a maximum generated force of about 5000–6000 N at low contraction. The goal of the biped project is to achieve a lightweight bipedal robot which is able to walk in a dynamically stable way, while exploiting the passive behaviour of the PPAMÕs in order to reduce energy consumption and control efforts. Fig. 3 shows the robot Lucy which weighs less than 30 kg and is 150 cm tall. The motion Fig. 1. Photograph of inflated state of the PPAM. 709 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 12000 1 bar 2 bar 3 bar 10000 Force (N) 8000 6000 4000 2000 0 0 5 10 15 20 25 30 35 40 Contraction (%) Fig. 2. Generated forces for the muscle used in the robot ‘‘Lucy’’ (N). Fig. 3. Photograph of Lucy. 45 710 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 of Lucy is restricted to the sagittal plane in order to avoid extra unnecessary complexity regarding control and design. Moreover, it has been shown [36] that for biped walking the dynamical effects in the lateral plane have a marginal influence on the dynamics in the sagittal plane. Key elements in the design phase were modularity and flexibility to be able to make changes to the robot configuration during the experimental process. This resulted in nearly the same mechanical and electronic configuration for each structural element such as lower leg, upper leg and body. One such structural element consists of two muscles positioned in an antagonistic setup, since an artificial muscle can only generate a pulling force. The muscles drive a joint via a pull rod and leverage mechanism which is dimensioned to meet the needs of a specific joint. This dimensioning process determines joint range and torque characteristics. The mechanical design of the leverage mechanism allows to make easy changes to these characteristics. Currently, the angle ranges for ankle, knee and hip are set at (30° to 30°), (15–65°) and (35° to 15°) respectively. Hereby the ankle angle is defined between a perpendicular direction to the foot and the lower leg, the knee angle between the lower leg and the upper leg, and the hip angle is defined between the upper leg and the upper body. The graph of Fig. 4 shows the generated knee torque values for each muscle of the antagonistic setup at different working pressures. The actual applied knee torque is calculated by subtracting the two generated torques. Each modular unit is controlled by a 16 bit Motorola micro-controller, type MC68HC916Y3. On these micro-controller platforms the local joint control (see Section 3.4) is implemented. The local control unitÕs main task is to control the pressure in both muscles, record their pressure values and the joint angle position. This angular position is captured with a HEDM6540 incremental encoder from Agilent Technologies which has 2000 pulses per revolution, resulting in one detectable flank each 0.045°. The pressure values are measured by micro-silicon absolute pressure 80 front muscle rear muscle 3 bar 70 Torque (Nm) 60 50 2 bar 40 30 1 bar 20 10 0 15 20 25 30 35 40 45 Knee angle (˚) 50 55 60 65 Fig. 4. Knee torque values to knee angle for both muscles of the knee joint at 1, 2 and 3 bar. B. Verrelst et al. / Mechatronics 15 (2005) 703–729 711 sensors, CPC100 from Honeywell, whose amplified signals are immediately converted into a 12 bit serial digital signal. The accuracy of the pressure measurement is approximately 20 mbar. One of the control philosophies intended for ‘‘Lucy’’ is to combine trajectory, control based on objective locomotion parameters [37,38], with the natural dynamics of the system. More specifically, the natural dynamics of the system will be adapted as a function of the objective parameters and the calculated joint trajectories. This implies that a tool is needed to change the natural frequency of the system on-line while an actuator should be able to track the desired trajectories. The pleated pneumatic artificial muscle, in an antagonistic setup, is an actuator able to track trajectories while its adaptable passive behaviour can be adjusted in order to change the dynamics by means of compliance variations in the different joints. So the overall control system of Lucy has actually three main tasks: firstly realtime generation of joint trajectories which ensure dynamic stable walking, secondly tracking of these generated trajectories, and at last adjusting the compliance in the joints in order to tune the natural dynamics associated with the desired robot motion. The main focus of this paper concerns the second task, focussing specifically on the single support phase. 3. Control architecture The control structure can be divided into two parts: a real-time joint trajectory generator and a trajectory tracking controller. The latter consists of a computed torque controller, a delta-p unit, a PI controller and a bang–bang pressure controller. Fig. 5 gives an overview of the applied control scheme. A trajectory generator provides polynomial joint trajectories while a computed torque controller calculates the required joint torques taking the complete dynamics of the robot into account. For each joint the delta-p unit translates the calculated torques into desired pressure levels for the two muscles of the antagonistic setup. Additionally, a correction Dppi, calculated by a local PI controller, provides a position feed-back to cope with modelling errors. And finally a bang–bang controller determines the necessary valve actions to set the correct pressures in the muscles. The trajectory generator, computed torque and delta-p units are implemented on a central PC since these controllers need a lot of computations. The PI controller and the bang–bang controller are locally implemented on the micro-controller units. Due to the use of a Windows operating system and the USB communication line, the refresh rate for the control calculations, implemented on the PC, is currently set to 2000 Hz, which is the same as the refresh rate of the local micro-controller units. The timing of the communication refresh rate is controlled by a USB micro-controller interface board (EZ-USB FX2 from Cypress Semiconductor), which transfers the serial data coming from the central PC to a 16 bit parallel bus connected to the different micro-controller units, on which the local control of the joints is implemented. These local micro-controllers ensure low-level, quasi real-time, control of the joints. And, in order to prevent control disturbance of missed torque calculations by the 712 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 Objective parameters . .. computer qdes, qdes, qdes q, q Computed torque T left ankle joint USB Delta-P unit θdes P2des −+ microcontrollers ∆p pi Local PI controller P1 P2 −+ + + Bang-bang Controller Valves 2 Actions Valves 1 Actions 2000 Hz sampling rate θ P1des Pm other joints 2000 Hz sampling rate Trajectory generator Robot q, q Fig. 5. Overview of the control architecture. central PC, the incoming data of the local units are buffered via dual ported RAM hardware, which is provided for each local micro-controller. So whenever the central PC does not succeed to perform the necessary calculations within the sampling time, the local control unit uses the previously sent data, which are stored in the dual ported RAM structure. One should also remark in the context of this refresh rate, that the delay time of the valves is about 1 ms, which suggests that the communication frequency of 2000 Hz is high enough. In the next sections the different parts of the control structure are discussed in more detail. 3.1. Trajectory generator In this section the trajectory generator is discussed briefly, a more elaborate discussion can be found in [37,38]. This control unit generates joint motion patterns based on two specific concepts, being the use of objective locomotion parameters, and exploitation of the natural upper body dynamics by manipulating the angular momentum equation. Objective locomotion parameters are average forward speed, step-length, step-height and intermediate foot-lift. These parameters should be calcu- B. Verrelst et al. / Mechatronics 15 (2005) 703–729 713 lated by a higher level path planning control unit, which is beyond the scope of this paper. It is important to mention that the trajectory generator ensures dynamically stable walking for a wide range of objective locomotion parameter combinations. Although this paper only reports on the performance of the controller during single support, the trajectory generator uses both single and double support phases to achieve an overall desired robot motion. During single support the trajectories of the leg joints, represented by polynomials, are planned in such a way that the upper body motion is naturally steered. As a consequence, practically no ankle torque is needed in the supporting leg. Only small ankle torques must be provided to compensate for modelling and approximation errors. Doing so, the zero moment point (ZMP) [39] is kept in the vicinity of the ankle joint and thus away from the supporting foot edges, resulting in a dynamically stable walking motion. The polynomials are established by manipulating the angular momentum equation to determine correct boundary conditions for the hip and upper body motion. A short double support phase is used to ensure the necessary start conditions for the next single support phase and switches the ZMP from the rear foot to the next supporting front foot. One of the most interesting aspects of this method is that it is based on fast converging iteration loops, requiring only a limited number of elementary calculations. The computation time needed for generating feasible trajectories is low, which makes this strategy suitable for real-time application on Lucy. 3.2. Computed torque controller The computed torque method [40], also called feedback linearization, is implemented to linearize the non-linear input–output relation for the mechanical dynamic equations describing the robot motion: _ q_ þ GðqÞ ¼ s DðqÞ€ q þ Cðq; qÞ ð2Þ With q a vector of independent Lagrange coordinates describing robot configura_ the centrifugal/coriolis matrix and G(q) the tion, D(q) the inertia matrix, Cðq; qÞ gravitational torque/force vector. The computed torque method determines the torque vector s, applied in each joint, by feeding forward the desired trajectory accelerations €qdes and by feeding back measured positions q and velocities q_ in order to cancel the non-linear centrifugal and gravitational terms. A secondary PD-feedback loop is added to influence control performance. This results in the following calculation: _ q_ þ Ge ðqÞ þ De ðqÞ½€ s ¼ C e ðq; qÞ qdes  K d ðq_  q_ des Þ  K p ðq  qdes Þ ð3Þ The matrices De, Ce and Ge contain the estimated values of the inertia, centrifugal and gravitational parameters. The feedback diagonal gain matrices Kd and Kp are tuned in order to make the mechanical system to behave as critically damped, if the model would be perfect. This secondary feedback loop also copes with unavoidable modelling errors of the mechanics such as parameter estimation errors, friction and the actuator tracking limitations. 714 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 3.3. Delta-p unit The computed torque unit produces the torques, required in each of the six joints, in order to track the calculated polynomials. On the other hand, the torques generated by each joint are determined by the pressures in the antagonistic muscle system. Therefore the delta-p unit is required to transform the calculated torques into desired pressure levels. For each muscle pair such a controller is provided and dimensioned according to its specific torque characteristic. The generated torque in an antagonistic setup with two muscles can be calculated with the kinematical model of the leverage and rod mechanism combined with the estimated force function of the muscles and the applied gauge pressures. This can be represented by the following calculation [34]: s ¼ s1  s2 ¼ p1 l21 r1 f1  p2 l22 r2 f2 ¼ p1 t1 ðhÞ  p2 t2 ðhÞ ð4Þ with p1 and p2 the applied gauge pressures in extensor and flexor muscles respectively, which have lengths l1 and l2, while f1 and f2 determine the estimated force function of each muscle. The kinematical transformation from forces to torques are represented by r1 and r2 which results, together with the muscle force characteristics, in the torque functions t1 and t2. These functions are determined by the choices made during the design phase and depend on the specific joint angle h. Eq. (4) is used to determine the two desired gauge pressures p1des and p2des for each muscle pair. These two pressures are generated starting from a mean pressure value pm while adding and subtracting a Dp value: p1des ¼ pm þ Dp ð5Þ p2des ¼ pm  Dp ð6Þ The mean value pm determines the joint stiffness and will be controlled in order to influence the natural dynamics of the system. In this paper the value of pm for each joint is set constant. A more elaborate discussion on the topic of changing stiffness and natural dynamics can be found in [41]. Combining Eqs. (5) and (6) with Eq. (4) allows to calculate the Dp value required to generate the torque originating from the computed torque module: Dp ¼ s þ pm ½ðt2 ðhÞ  t1 ðhÞÞ t2 ðhÞ þ t1 ðhÞ ð7Þ This unit is actually a feed-forward calculation from torque to pressure level and uses estimated values of the muscle force function and estimated kinematical data of the pull rod mechanism. The force functions of the muscles can be taken from the mathematical model as described in [34] or, as is the case for this paper, be measured with static force tests. The force to contraction curve shows a hysteresis which can result in a non-negligible force estimation error up to 5%. The extra local feedback PI controller discussed in the next section can again cope with these inaccuracies. B. Verrelst et al. / Mechatronics 15 (2005) 703–729 715 3.4. Local PI and bang–bang pressure controller As was mentioned in Sections 3.2 and 3.3, a local PI controller is implemented on the different micro-controller units. This controller is directly acting at pressure level by adding and subtracting an additional Dppi in Eqs. (5) and (6). The value of Dppi is calculated by a proportional and integral feedback structure on joint angle position: X ðh  hdes ÞDtlocal Dppi ¼ K plocal ðh  hdes Þ þ K ilocal ð8Þ with K plocal and K ilocal the local feedback gains, h the measured joint angle and hdes the desired joint angle originating from the calculated reference functions. The sampling time Dtlocal is set at 0.5 ms which is a result of the refresh rate of 2000 Hz for the calculations on the micro-controller units. In order to realize a lightweight rapid and accurate pressure control, fast switching on–off valves are used. The pneumatic solenoid valve 821 2/2 NC made by Matrix weighs only 25 g. With their reported opening times of about 1 ms and flow rate of 180 Std.l/min, they are about the fastest switching valves of that flow rate currently available. To pressurize and depressurize the muscles, which have a varying volume up to 400 ml, a number of these small on–off valves are placed in parallel. Obviously the more valves used, the higher the electric power consumption, the price and also the weight will be. Simulations of a pressure control of a constant volume lead to the compromise of two inlet and four outlet valves. The different number between inlet and outlet comes from the asymmetric pressure conditions between inlet and outlet combined with the aim to create comparable muscleÕs inflation and deflation times. The pressure control in a volume is achieved with a bang–bang controller with various reaction levels depending on the pressure error. If this error is large two inlet or four outlet valves, depending on the sign of the error, are opened. If this error is smaller only one valve is switched and when the error is within reasonable limits no action is taken, leaving the muscle closed. The principle of this control scheme is depicted in Fig. 6. The absolute values of reaction levels currently are set at 20, 25 and 60 mbar. More detailed information on the valve system can be found in [42]. Fig. 6. Multi-level bang–bang control scheme. 716 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 4. Simulation model In this section a simulation model of the biped ‘‘Lucy’’ is discussed. This model is a full hybrid simulation with mechanics and thermodynamic processes combined in one set of differential equations. Doing so includes the muscle actuator characteristics which allows to study the limitations in the motion of the robot. Furthermore it is possible to investigate if dynamic balance as supposed by the trajectory generator is still ensured under actuator and hardware limitations and parameter inaccuracies. Finally, this model is used to design and evaluate the proposed control structure and can give good understanding of the working of muscle actuator/valve system in legged robots. 4.1. Mechanics The biped model during a single support phase is shown in Fig. 7, with Gi, mi and Ii respectively the center of gravity, mass and moment of inertia around Gi of each link and Ji the rotation axis between two links. The inertial and geometrical parameters for the simulation model are summarized in Table 1 with li the length of link G3,m3,I3 θ3 J4 J3 G4,m4,I4 G2,m2,I2 θ4 θ2 J5 G5,m5,I5 θ5 J6 G6,m6,I6 J2 θ6 Y G1,m1,I1 θ1 Z J1 X Fig. 7. Model of the biped in single support. Table 1 Inertial parameters of ‘‘Lucy’’ i li (m) JiGi (m) mi (kg) Ii (kg m2) 1 2 3 4 5 6 0.45 0.45 0.45 0.45 0.45 0.3 0.260 0.261 0.2 0.262 0.258 0.056 3.61 3.69 18.0 3.66 3.53 1.13 0.060 0.062 0.6 0.06 0.058 0.005 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 717 i. Except for the upper-body parameters, all values are obtained by measurement. For the upper-body different values are considered in order to simulate a possible payload which can be carried by the robot. The mechanics are described by Eq. (2), as described in Section 3.2, written for all links except for the stance foot. The position vector q is given by the following absolute coordinates: q ¼ ½ h1 h2 h3 h4 h5 T h6  ð9Þ The origin of the X, Y-coordinate system is located at the ankle joint of the supporting _ the equations of motion can be written as: foot. Introducing x ¼ q, DðqÞx_ þ Cðq; xÞx þ GðqÞ ¼ s ð10Þ Since the inertia matrix D(q) is symmetric and positive definite, it can be inverted: ( 1 x_ ¼ DðqÞ ½s  Cðq; xÞ  GðqÞ ð11Þ q_ ¼ x Eq. (11) represents a set of 12 first order differential equations for which the torque s is dependent on the angular positions q and the pressure values in the muscles of all joints (Eq. (4)). 4.2. Thermodynamics The pressure inside a muscle is influenced by the volume changes resulting from the variation of the joint angle and by the orifice flows through the valves which have been activated by the bang–bang pressure controller. Assuming a polytropic thermodynamic process, and assuming that the compressed air inside each muscle behaves as a perfect gas, the first law of thermodynamics, while neglecting the fluidÕs kinetic and potential energy, can be rewritten for each muscle of the antagonistic setup in the following differential form [35]: p_ i ¼ n _ _ in _ ex ðrT sup airi  rT airi m airi  ðP atm þ p i ÞV i Þ air m Vi ð12Þ with r the dry air gas constant and n the polytropic exponent. T sup air is the temperature of the supply air and T airi the temperature in muscle i. The volume of muscle i is represented by Vi and Patm is the atmospheric pressure. The total orifice flow through _ ex the opened inlet valves and exhaust valves of muscle i are given by m_ in airi and m airi respectively. The latter two can be calculated with following equations which represents the normalized approximation of a flow through a valve defined by the International Standard ISO6358 [43]: m_ air sffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 293 P d =P u  b Pd 1 ¼ CP u q0 if Pb T uair 1b Pu ð13Þ 718 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 m_ air sffiffiffiffiffiffiffiffi 293 Pd 6b if ¼ CP u q0 T uair Pu ð14Þ with q0 the air density at standard conditions. C and b are two flow coefficients characterizing the valve. Both coefficients have been experimentally determined for the used matrix valves, which resulted in C = 22 Std.l/min bar and b = 0.16. Pu and Pd are the upstream and downstream pressures, while T uair is the upstream temperature. When choking occurs Eq. (14) is valid, and in the other case Eq. (13) is used. Once the actions (opening of the valves) for the different inlet and exhaust valves are known, all the orifice flows can be calculated in order to be substituted in (12). The temperature in the muscle is calculated with the perfect gas law: T airi ¼ PV mairi r ð15Þ Here the total airmass mairi is given by integration of the net mass flow entering muscle i: _ ex m_ airi ¼ m_ in airi  m airi ð16Þ The volumes and their derivatives to time are given by kinematical expressions in function of the joint angle and joint angle velocity. These functions are determined by the muscle dimensions as described in [35] and by the geometry of the pull rod system. The link at torque level between the mechanical equations of motion and these thermodynamic differential equation systems is provided with Eq. (4) which characterizes joint torque in function of pressures. 4.3. Complete simulation model In Fig. 8 an overview is given of the complete simulation model. The kernel of this simulation model is based on three different equation blocks, as depicted in the center of Fig. 8. The 12 first order differential equations (11) describe the motion of the robot during single support. The thermodynamics of each joint are characterized by four first order differential equations on pressure (12) and airmass (16). This leaves a total set of 24 differential equations for the thermodynamic differential equation block. Finally, another 12 thermodynamic state equations, the perfect gas law (15), complete the set. The antagonistic muscle model block creates the link between the mechanics and the thermodynamics by calculating the torque for each joint with the pressure information of the thermodynamic block. Therefore it needs angle information from the integrated equations of motion. This information allows to calculate the contraction of each muscle within the antagonistic setup, while using the kinematic data of the pull-rod mechanism of the specific joint. With the contraction values, the linear forces (1) of the two muscles can be calculated in order to determine the applied torque with Eq. (4). Additionally, to determine the pressure changes in the 719 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 Runge-Kutta numerical integration integration timestep 50 microseconds Psupply Tsupply Equations of motion qi , ωi τi . i . i m inairj / m ex airj Valve system model p i j Thermodynamic differential equations i 1 i 2 Thermodynamic state equations Valve actions Controller selection Valve controls Antagonistic muscle model joint parameters T Delay observer . p /p i airj t . V1i / V2i V1i / V2i i Kinematic joint data q , ω ,p,t objective parameters Control unit SS/DS sampling time 500 microseconds control parameters Fig. 8. Structure of the complete simulation model. thermodynamic differential equation block, muscle volume and volume changes are calculated. For the volume changes angular velocity information is needed from the integrated equations of motion. The valve system model block determines for each muscle the mass airflow rates ((13) or (14)), depending on the actual pressure and temperature in the muscle and the action taken by the valves. This action is determined by the valve control signals of the control unit. But these signals first pass through the delay observer, which needs the time instant of the integrator to determine whether the valve may be opened or not. Hereby a valve delay of 1 ms is taken into account. The differential equations are numerically integrated using a forth order Runge– kutta method with an integration time step of 50 ls, which is 10 times less than the sample time of the control unit. In order to evaluate robustness of the controller with respect to parameter estimation, two systematic errors are introduced. The computed torque calculates with deviations on the inertia parameters: 5% for center of gravity and mass and 10% for the inertia of each link. For the delta-p unit the reported 5% for the hysteresis on both force functions of the antagonistic set-up is taken into account. 5. Results and discussion In this section a simulation is shown for one single support phase. The objective parameters are chosen as follows: step-length 0.15 m, step-height 0, footlift 0.02 m and the mean forward speed 0.2 m/s, which results in a calculated single support stance time of about 0.58 s. 720 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 The control strategy for this biped is not to achieve high precision tracking but is focused on implementing flexible actuator characteristics in order to influence energy consumption. So the design of the controllers is based on a compromise between tracking error and control activity while ensuring the overall robot stability and attaining the objective parameter values. In Fig. 9 the absolute angle h2 and the angular velocity for the upper-leg of the stance leg is given. The graphs show errors at position and velocity level which are a consequence of the nature of the pneumatic drive unit and its control principle. The pneumatics are characterized by pressure courses in both muscle of each joint. Figs. 10 and 11 depict desired and measured pressure for the muscles in the front and the back of the knee joint together with the respective valve actions. Note that in these figures a closed muscle is represented by a horizontal line depicted at the same level of the initial pressure while a small peak upwards represents one opened inlet valve, a small peak downwards one opened exhaust valve and the larger peaks represent two opened inlet or four opened outlet valves. The desired pressures are calculated by the delta-p unit and corrected by the local PI-controller. For this simulation the mean pressure pm for all joints is taken at 2 bar, consequently the sum of the pressures in the Graphs 10 and 11 is always approximately 4 bar. Due to the discrete levels of the bang–bang pressure controller and its death zone and due to the opening and closing times of the valves the latter are not switching continuously. This is important towards energy consumption but of course has its reflection on the tracking error. The torque generated by these two pressure courses for the knee joint is given in Fig. 12. This figure includes the theoretical feed-forward torque calculated with the Lagrange equations of the mechanical robot model. The pneumatics and parameter estimation errors introduce deviations between calculated and actually applied torques. Fig. 9. Absolute angle h2 (°) and angular velocity (°/s). 721 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 Pressure front muscle (bar) 3.6 3.4 3.2 3.0 2.8 actual desired 2.6 2.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Fig. 10. Desired and actual pressure (bar) in the front muscle for the knee joint of the stance leg. Pressure rear muscle (bar) 1.4 1.2 1.0 0.8 actual desired 0.6 0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Fig. 11. Desired and actual pressure (bar) in the rear muscle for the knee joint of the stance leg. One of the important strategies of the trajectory generator is to achieve natural upper-body motion by choosing appropriate leg link trajectories. In fact the hip moves in such a way that the upper-body rotation is kept small. This can be seen in Fig. 13 where the absolute angle and angular velocity of the upper body are drawn. The body rotates naturally with small amplitude around the upward body posture. During the next double support phase, which is not discussed in this paper, the remaining upper body angle deviation between initial and end state of the single support phase will be compensated. 722 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 40 Joint torque (Nm) 35 30 25 20 torque (pneumatic tracking) torque (perfect tracking) 15 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) Fig. 12. Torque values in the knee of the stance leg with perfect tracking compared to tracking with the pneumatic system. Fig. 13. Absolute angle h3 (°) and angular velocity (°/s). As a result of this upper-body strategy, only a reduced torque has to be exerted in the ankle joint of the stance leg in order to achieve this upper-body motion. Fig. 14 shows both theoretically required and actually applied torque in the ankle joint. Again deviations between these two exist due to the pneumatic actuator system and parameter deviations, but torque values are still limited. A small ankle torque has an important reflection on the ZMP which is depicted in Fig. 15. Due to the low ankle torque the theoretical ZMP would only oscillate with small amplitude around the ankle point. The actual ZMP moves in a similar way and only deviates B. Verrelst et al. / Mechatronics 15 (2005) 703–729 723 Fig. 14. Torque values in the ankle of the stance leg with perfect tracking compared to tracking with the pneumatic system. Fig. 15. Zero moment point position with perfect tracking compared to tracking with the pneumatic system. with a maximum amplitude of 3 cm away from the ankle joint. A zero moment point that reaches the physical boundary of the supporting foot during single support, makes the robot tip over in the sagittal plane which has to be prevented in order to ensure postural stability. The feet of the robot ‘‘Lucy’’ have respective lengths of 20 cm to the tip and 10 cm to the heel of the foot, so the overall stability is guaranteed even with the introduced parameter deviations and imperfect pneumatic tracking system. 724 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 6. Preliminary tracking experiments Currently, the feed-forward trajectory tracking algorithm, as proposed in this article, has been programmed and tested on the real robot only during walking motion while the robot is suspended in the air. A full dynamic walking motion cannot be achieved since double support control strategies are not yet implemented. Moreover, a treadmill is currently being installed to perform continuous walking experiments. In order to evaluate the tracking performance, the proposed feed-forward structure has been implemented on the robot while it mimics essential walking motion, but with both feet in the air. The speed of the motion, if the robot were walking on level ground, is about 0.5 km/h. A video of this motion can be seen at the following internet address: http://lucy.vub.ac.be/movies/lucy_airwalk.WMV. Fig. 16 gives the desired as well as measured angles for the ankle, the knee and the hip joint of one leg. The presented graphs depict two walking cycles and clearly show the effectiveness of the control strategy. Note that the presented graphs include two double support phases and four single support phases. Tracking errors are bounded within a maximum of one to two degrees, which is a good achievement for a pneumatic system in a dynamic tracking application. Moreover, the simulation results discussed in Section 5 showed that positions errors of comparable magnitude do not jeopardize global stability of the robot. Figs. 17 and 18 depict desired and measured pressure values for the knee. Both pressure courses of front and rear knee muscle are shown for two walking cycles. The mean pressure pm is set at 2.5 bar, which results in a sum of both pressures of approximately 5 bar. Additionally, both graphs show the valve activity which is for this movement restricted to switching of only one inlet or outlet valve. In Fig. 19 Fig. 16. Desired and measured angles (°) for the ankle, knee and hip joint. B. Verrelst et al. / Mechatronics 15 (2005) 703–729 725 Fig. 17. Desired and measured pressure (bar) in the front muscle for the knee joint. Fig. 18. Desired and measured pressure (bar) in the rear muscle for the knee joint. a detail of the pressure course and valve switching for the front muscle in the knee is given for the time interval 1–2 s. These experimental results show promising results of the pneumatic tracking system. In combination with the real-time trajectory generator, the tracking system should lead to a dynamically stable walking motion of the robot ‘‘Lucy’’. The next step in the experimental process will be the implementation of a double support phase tracking control strategy and the provision of a speed controlled treadmill to perform continuous walking experiments. 726 B. Verrelst et al. / Mechatronics 15 (2005) 703–729 Fig. 19. Detail of desired and measured pressure (bar) in the front muscle for the knee joint. 7. Conclusions This paper discussed in detail the control architecture of the biped ‘‘Lucy’’ which is actuated with pleated pneumatic artificial muscles. Although not widely used in the community of legged robotics, these devices can give an interesting alternative for electrical drives. They have a high power to weight ratio, can be directly coupled to steer a joint, and their inherent compliance reduce impact shocks. The adaptation of the compliance can be exploited to influence the natural dynamics of the robot in order to reduce energy consumption and control efforts. The discussion of the control architecture focused on the tracking controller which was divided into a computed torque controller, a delta-p unit, a PI position controller and a pressure bang–bang controller. The formulation of the computed torque controller is given for the single support phase only and calculates the necessary joint torques required to track the desired positions. The latter are generated in real-time by the trajectory generator which is based on two specific concepts: the use of objective locomotion parameters and exploitation of the upper-body dynamics by manipulating the angular momentum equation. The control strategy was evaluated with a hybrid simulation model of the biped. This model combined formulations of the thermodynamic processes, taking place in the actuators, with the standard Lagrange representation of the robot dynamics. Reported hardware limitations such as valve delays and sampling times, observed on the real robot, where taken into account in the simulation model, as well as some expected parameters estimation errors. Results show the effectiveness of the tracking controller and clearly explain the nature of the pneumatic pressure control unit and its implications on tracking precision. Although not perfect, this tracking task does not undermine the main scope of assuring global dynamic stability. This was B. Verrelst et al. / Mechatronics 15 (2005) 703–729 727 represented by the location of the ZMP which was situated in the vicinity of the ankle joint mainly due to the low ankle torque values applied on the support leg. Standard PID control tracking techniques with a quasi-static trajectory generator have been implemented on the real robot to evaluate and debug the hardware. This resulted in slow real walking patterns. Currently, the feed-forward trajectory tracking algorithm, as proposed in this article, has been programmed and tested on the real robot during walking motion just while the robot is suspended in the air. This paper showed promising preliminary results of the tracking performance. Some videos of these achievements can be seen at the website: http://lucy.vub.ac.be. The next step is to extend the control architecture to double support phase and implement the proposed elaborate control structure in the real model. For practical walking test a treadmill is currently being developed, which is required to guide the robot during continuous walking motion. And in the near future strategies to exploit the adaptable compliance will be formulated and tested. The fast and accurate pressure control and the feed-forward structure, which combines mechanical and pneumatic actuator modelling on one hand, and the proper design of the pleated pneumatic artificial muscle on the other hand, results in a pneumatic actuation system suited for dynamic control. Most robotic applications which implement pneumatics are restricted to slow and quasi static motions. The results shown in this paper reveal a promising future for these devices to be used in dynamic applications such as walking robots. It is therefore important not only to design and investigate a trajectory tracking controller, but also study a trajectory generator which tackles the complex task concerning dynamic stability of biped walking. Moreover, the inherent compliance should be exploited in order to tune the natural behaviour of the mechanical system and be beneficial during impact, which becomes crucial when walking speed are increased. 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