Biol. Cybern. 90, 368–375 (2004) DOI 10.1007/s00422-004-0484-4 © Springer-Verlag 2004 A model of force and impedance in human arm movements K. P. Tee1 , E. Burdet1,2 , C. M. Chew1 , T. E. Milner3 1 2 3 Department of Mechanical Engineering, National University of Singapore Division of Bioengineering, National University of Singapore School of Kinesiology, Simon Fraser University, Canada Received: 24 July 2003 / Accepted: 21 April 2004 / Published online: 14 June 2004 Abstract. This paper describes a simple computational model of joint torque and impedance in human arm movements that can be used to simulate three-dimensional movements of the (redundant) arm or leg and to design the control of robots and human–machine interfaces. This model, based on recent physiological findings, assumes that (1) the central nervous system learns the force and impedance to perform a task successfully in a given stable or unstable dynamic environment and (2) stiffness is linearly related to the magnitude of the joint torque and increased to compensate for environment instability. Comparison with existing data shows that this simple model is able to predict impedance geometry well. Keywords: Motor adaptation – Impedance – Force – Stable and unstable interactions 1 Introduction Most activities that we perform with our hands involve interaction with the environment. This interaction imposes forces on the hand and can also destabilize motion. However, humans have excellent capabilities to manipulate objects. This means that the central nervous system (CNS) is able to adapt to various task dynamics. For example, one may have difficulty in opening a door for the first time due to unknown friction. However, after one or two trials the appropriate force will be learned, and one will open the door without difficulty and even without thinking about it. Many tasks performed with tools are inherently unstable (Rancourt and Hogan 2001) and consequently require the acquisition of additional skills, because in unstable tasks different initial conditions, neuromotor noise (Schmidt et al. 1979; Slifkin and Newell 1999), or any small external perturbation can lead to inconsistent Correspondence to: E. Burdet (e-mail: e.burdet@ieee.org, http://guppy.mpe.nus.edu.sg/∼eburdet) and unsuccessful performance. To learn more about human motor adaptation, recent works have investigated the adaptation to stable (Shadmehr and Mussa-Ivaldi 1994; Shadmehr and Holcomb 1997; Conditt and Mussa-Ivaldi 1997) and unstable (Burdet et al. 2001; Franklin et al. 2003b; Osu et al. 2003; Franklin et al. 2003c) interactions produced by a haptic interface. Understanding how humans interact with the environment can provide insight into the neural mechanisms of motor learning and adaptation with potential application to computer animation of human motion and neuromotor rehabilitation and to the development of robots collaborating with human operators. On one hand, it may be advantageous to use humanlike control strategies for improving robot control, in particular with regard to safety (Bicchi et al. 2001). On the other hand, there have been recent significant advances in medical robotics (Surgical robots 2004) and teleoperation and micro/nanotechnology (Zhang et al. 2002), all of which rely on haptic interfaces to allow the human operator to feel and manipulate objects not directly accessible to the hand. The safety and performance of such systems critically depend on motion stability, i.e., the coupled stability of the arm interacting with the environment (Colgate and Hogan 1988). To adequately control these systems, it would be useful to know the mechanical impedance, i.e., the resistance to infinitesimal perturbations applied at the hand. Mechanical impedance of the human arm can be evaluated from the restoring force to slight perturbations imposed in static positions (Mussa-Ivaldi et al. 1985) or during movement (Bennett et al. 1992; Bennett 1993; Milner 1993; Gomi and Kawato 1997; Burdet et al. 2000). It is possible to measure the impedance in movements involving a given stable or unstable interaction (Burdet et al. 1999, 2001; Franklin et al. 2003b). However, this requires many movements, and it would be more useful to have a compact model to describe the force and impedance in every dynamic interaction. Impedance was shown to depend on position (Mussa-Ivaldi et al. 1985), force (Gomi and Osu 1998; Perreault et al. 2001), and instability (Burdet et al. 2001), but no comprehensive model has been proposed so far. This paper introduces a simple model of 369 force and impedance formulated in joint space, provides comparisons with previous measurements that show its predictive power, and discusses its relevance and the extent of its validity. 2 Model The model introduced in this section describes how the arm force and impedance, necessary to simulate arm motion in a given environment, can be computed from the dynamic characteristics of the environment. The following description uses bold vectors v and matrices M and italic scalars s. 2.1 Joint torque The joint torque τ produced by muscles to perform a learned task consists of two components, viz. the torque to compensate for the external force FE applied on the hand and the torque τ B necessary to move the limbs: τ = −J(q)T FE + τ B , (1)
 Nxi where J(q) ≡ Nqj is the (position-dependent) Jacobian of the transformation between Cartesian and joint space, i.e., the external force is transformed into joint torque using the Jacobian. q is the vector of joint angles. How can τ B be evaluated? The main forces arising when moving the bones and flesh are the inertia of the corresponding rigidbody dynamics τ B , elasticity due to muscles, tendons, etc., and reflexes. Psychophysical experiments (Shadmehr and Mussa-Ivaldi 1994; Lackner and Dizio 1994; Conditt and Mussa-Ivaldi 1997; Franklin et al. 2003b) suggest that, for movements repeated in a novel dynamic environment, the CNS learns to compensate for the predictable forces exerted on the arm during movement by forming an internal inverse dynamics model of the task (Kawato 1999). Elastic forces and reflex feedback forces, arising from trajectory perturbations, decrease as the CNS learns to compensate for the perturbations (Milner and Cloutier 1993; Thoroughman and Shadmehr 1999; Franklin et al. 2003c), and we assume that they become negligible after learning. Thus we assume that τ B corresponds to the rigid-body dynamics (De Wit et al. 1996) described by τ B = M(q)q̈ + C(q, q̇)q̇ + G(q) , (2) where q̇ and q̈ are the joint velocity and acceleration vectors, respectively, M(q) is the (position-dependent) mass matrix, C(q, q̇)q̇ the Coriolis and centrifugal velocity dependent forces, and G(q) the gravity term. The force due to gravity can be estimated from static measurements and the inertia from rapidly accelerating movements. A jointvelocity-dependent term can be added to this equation to account for joint damping. 2.2 Impedance If one considers that the neural control of the human arm (expressed as joint torque τ ) depends on position q, velocity q̇, acceleration q̈, and muscle activation u(q,q̇) : τ = τ (q, q̇, q̈, u), then the mechanical impedance (defined as resistance to perturbations at the hand) also depends on these variables: δτ = K δq + D δ q̇ + M δ q̈ , (3) 
where K = NNqτji + NNuτik NNuqjk corresponds to joint stiffness,

 D = NNq̇τji + NNuτik NNuq̇jk represents damping, and M = NNq̈τji is the inertia. Current methods for measuring stiffness and damping cannot isolate the activation-dependent parts, i.e., reflex feedback, from muscle intrinsic viscoelastic properties (Burdet et al. 2000). Thus, measurements normally combine both components, corresponding to the definitions of K and D above. Damping is difficult to estimate in the multijoint case, particularly as its contribution to the impedance is small. We will assume that it is roughly proportional to stiffness, corresponding to results obtained under isometric conditions, i.e., in static postures (Tsuji et al. 1995). More precisely, the viscosity in joint space is assumed to be related nonlinearly to joint stiffness and depend on velocity such that 0.42 K. D=  (q̇T q̇ + 1) (4) This gives relatively higher damping at the start and end of movements when the velocity is low. How can stiffness be modeled? We first assume, in accordance with experimental results, that (in joint space) stiffness K does not depend explicitly on position or velocity. In a stable static interaction, joint stiffness is linearly correlated (Gomi and Osu 1998; Perreault et al. 2001) with torque magnitude |τ | = (|τ 1 |, |τ 2 |). The results of (Burdet et al. 1999) and (Franklin et al. 2003b) suggest that a similar linear relation holds during motion. Therefore, we assume that joint stiffness Kq (|τ |) depends linearly on the magnitude of the torque |τ | produced at the joints, which is computed from (1). This means that with our model, measurement of stiffness in static positions is sufficient to predict stiffness during motion performed during interaction with a dynamic environment. From (1), the stiffness geometry will depend on the force exerted on the hand by the environment, on the torque necessary to move the arm and on the arm geometry. Finally, to account for the increase of stiffness observed (after learning) in unstable interactions (Burdet et al. 2001), we assume an additional increase of stiffness Ki ≥ 0, represented in Cartesian space, which is independent of torque and opposes the environment instability. In summary, joint stiffness after learning is expressed as K = Kq (|τ FF |) + JT Ki J. (5) Our recent results (Osu et al. 2003; Franklin et al. 2003c) suggest that force and impedance are controlled using two separate processes corresponding to reciprocal activation of agonist–antagonist muscle groups and their coactivation to succeed in unstable tasks such as carving. In this view, the torque-dependent stiffness term Kq (|τ FF |) corresponds to the reciprocal activation process and JT Ki J to coactivation. 370 Fig. 1. Scheme of arm control model. The feedforward τ FF corresponds to the learned force and feedback τ FB to the muscle elastic property (including muscle material properties and reflex feedback) 3 Simulations and results 3.1 Static stiffness and dependence on force The above model is valid for movements of the arm in the three-dimensional space. However, the simulations of this paper are restricted to horizontal two-joint arm movements for which we can compare the results with experimental data. Torque-dependent joint stiffness Kq is modeled using the mean linear relation for five adult subjects found by Gomi and Osu (1998):   10.8 + 3.18 |τ1 | 2.83 + 2.15 |τ2 | Nm/rad , (8) Kq = 2.51 + 2.34 |τ2 | 8.67 + 6.18 |τ2 | Fig. 2. Workspace of simulated movement tasks. The grey arrow represents transverse movement from (−0.2, 0.45) to (0.2, 0.45) m and the black arrow represents longitudinal movement from (0, 0.3) to (0, 0.55) m. Cartesian position coordinates are relative to the shoulder 2.3 Computing movements in interaction with a known environment The above model of force and impedance enables simulation of a movement of the (possibly redundant) arm or leg along a planned trajectory qp (t) in an environment characterized by force and impedance. First, the torque is computed using (1) and (2), then impedance with (5) and (4). Impedance is used to compute the effect of novel dynamics or unexpected disturbances △τ . The interaction dynamics are described by −J(q)T FE + τ B + △τ = τ = τ FF + τ FB , (6) where τ FF corresponds to a feedforward estimate of the interaction force, which has been learned, and the feedback component τ FB = K(qp − q) + D(q̇p − q̇) , (7) corresponds to the viscoelastic properties of the arm (Mussa-Ivaldi et al. 1985; Bennett et al. 1992; Bennett 1993; Milner 1993; Gomi and Kawato 1997) and produces a restoring force toward the planned trajectory (Fig. 1). The movement kinematics are obtained by numerical integration from (6). where subscript “1” stands for shoulder and “2” for elbow (Fig. 2). This estimate comprises both muscle elasticity and reflex feedback. Let the shoulder joint be the origin of the workspace coordinates (x, y)T in the horizontal plane, with x and y expressed in meters. Using (5) with τ FF = FE ≡ 0, Ki ≡ 0, static hand stiffness values were computed at five locations (Fig. 3a): (−0.2, 0.45), (−0.13, 0.45), (0, 0.45), (0.13, 0.45), and (0.2, 0.45) m. To visualize the elastic restoring force corresponding to a unit displacement of e ,e∈ the hand, we plotted stiffness ellipses, defined by {Kx |e| 2 ℜ }. In the nonredundant case of the present simulations, Kx can be computed from (6) as follows:   dJT Kx = J−T K − F J−1 , (9) dq which stems from the following derivation:      Nτi Nxk dJT T NFi K≡ F =J + Nqj N xk Nqj dq dJT F. ≡ J T Kx J + dq (10) The principal axes of the ellipse correspond to the singular values of the stiffness matrix Kx . The static ellipses were elongated with an orientation approximately aligned with the radial axis from the shoulder joint (Flash and MussaIvaldi 1990). Comparison with measured stiffness (Gomi and Kawato 1997) shows a close match in the size, shape, and orientation of the hand stiffness ellipses at similar postures. Note that the ellipses were slightly more elongated in the model simulation than in the data, as they correspond to different “subjects”. The parameters used in (8) are the mean parameter values of several subjects, not the specific parameter values of the subject whose data are being compared. 371 A B movement posture model 0.1 m experiment Fig. 3a, b. Stiffness ellipses for transverse movements and postures. Static stiffness (a) and dynamic stiffness (b) predicted by the model are compared with that measured in Gomi and Kawato (1997) (reproduced with permission). Start and end positions as well as duration of movement are similar in both model and measured data (Sect. 3.2) 100 N/m 3 4 3 2 A 5 4 1 B 2 5 1 200 N/m 8 6 7 6 8 7 The stiffness dependence on endpoint force was examined for two postures, for different magnitudes and directions of endpoint force (Fig. 4). The force magnitudes were 5, 10, 15, and 20 N and the eight directions span 360◦ at equal intervals of 45◦ . At both locations, the stiffness ellipse enlarged as the force was increased. The shape and orientation remained unchanged for a given direction but varied as the direction changed. When the endpoint force produced by the arm was in direction 3 or 4, the ellipse became increasingly narrow as the force increased and had a significantly different shape from that due to an endpoint force in the opposite direction (7 or 8, respectively). This corresponds to the features found in the experimental results of Gomi and Osu (1998). The above results show that the model is able to predict stiffness geometry well at static positions. Fig. 4a, b. Predicted postural endpoint stiffness for different endpoint force magnitudes and directions at (0, 0.35) m (a) and (0, 0.45) m (b) relative to the shoulder joint. The endpoint force generated by the arm has magnitudes of 5, 10, 15, and 20 N increasing outwards in eight directions: (−1, 0), (−1, −1), (0, −1), (1, −1), (1, 0), (1, 1), (0, 1), and (−1, 1), corresponding to the indices 1–8, respectively. The center ellipse is the stiffness when the endpoint force is zero Table 1. Anthropometrical data for arm segments Upper arm Forearm Mass Length Center of mass (kg) (m) from proximal joint (m) Mass moment of inertia (kg m2 ) 1.93 1.52 0.0141 0.0188 0.31 0.34 0.165 0.19 We simulated the experiments of Gomi and Kawato (1997), who studied horizontal movements at shoulder height with the hand supported such that the influence of gravity could be neglected. We chose the transverse movement from (−0.2, 0.45) to (0.2, 0.45) m and the longitudinal movement from (0, 0.55) to (0, 0.3) m, similar to the experimental movements. We assumed that the hand followed a minimal-jerk planned trajectory in Cartesian space (Flash and Hogan 1985): 3.2 Dynamic stiffness x (t) = A (6tn5 − 15tn4 + 10tn3 ), tn = t/T , To simulate the motion dynamics, the arm is modeled as a two-link rigid mechanical structure with anthropometrical data for the segments shown in Table 1. where x is the tangential displacement, A the movement amplitude, t the time, and T = 1s the duration of movement. The planned trajectory in Cartesian space is (11) 372 B A posture movement experiment model 100 [N/m] 100 [N/m] 0.1 [m] 0.1 [m] Fig. 5a, b. Stiffness ellipses for longitudinal movements and postures. a Simulated dynamic stiffness during outward and inward longitudinal movements (solid ellipses) and static stiffness at corresponding positions lying along the trajectory (dotted ellipses). Directions of movement are denoted by the arrows. b Comparison of simulated stiffness and stiffness measured by Gomi and Kawato (1997) for inward longitudinal movement (reproduced with permission). Start and end positions, as well as duration of movement, are similar in both model and measured data (Sect. 3.2) x(t) = x0 + (0, x )T T x(t) = x0 + (x , 0) for longitudinal movement in y for transverse movement in x , (12) where x0 is the start position. Joint angles q = (q1 , q2 )T were then obtained from Cartesian position x = (x1 , x2 )T via an inverse kinematics transformation (De Wit et al. 1996). The rigid body dynamics τ B were computed along the planned trajectory using (2). The external force FE acting on the hand is given by the dynamics of the haptic interface used in (Gomi and Kawato 1997; Burdet et al. 1999, 2001; Franklin et al. 2003c), with parameters identified as FE = ME ẍ + Dd ẋ + tanh(200 Ds ẋ) , (13) where   1.516 0 Ns2 /m, ME = 0 1.404   10.247 0 Dd = Ns/m, 0 7.592   0.102 0 Ds = Ns/m, 0 0.356 where ẍ and ẋ represent Cartesian acceleration and velocity, respectively. To show the effect of movement and the dynamics of the haptic interface, we computed stiffness ellipses during movement and compared them with the corresponding static ellipses at the same hand positions (Figs. 3 and 5). Snapshots of the stiffness ellipses were taken at regular temporal intervals of 100 ms. For both transverse and longitudinal movements, stiffness was higher as compared to postural stiffness, since joint torque was larger in magnitude because of the greater muscle force necessary to move the limb segments. In the transverse movement, there was a large generalized increase of stiffness, similar to what was also observed in measured movement data (Gomi and Kawato 1997). In the longitudinal movement toward the body, the increase in stiffness was slight and mainly in the direction of movement. As shown in Fig. 5b, the size, shape, and orientation of stiffness ellipses again correspond well to experimental data from similar movements (Gomi and Kawato 1997). The middle column of Fig. 5a shows the model prediction for a movement away from the body (for which no experimental data have yet been published). To visualize how stiffness magnitude and geometry vary with movement speed, stiffness ellipses were plotted for longitudinal and transverse movements at different speeds (Fig. 6). In contrast to Fig. 6, in which the interaction with the PFM was considered for comparison with data (Gomi and Kawato 1997), no external dynamics were considered during these movements. Movements with time scaling factors of 2, 1, and 0.5 were compared, corresponding to durations of 0.5, 1, and 2 s, respectively. Stiffness ellipses are plotted for nine equal time intervals. The start and end positions for the movements are the same as before. As we see in Fig. 6, an increase of movement speed, i.e., of the force to move the limbs, produced an isomorphic stiffness enlargement. The increase in size was most marked in the acceleration and deceleration phases of the movement. There was no change in the shape and orientation of stiffness among different movement speeds for either movement. 3.3 Adaptation to stable and unstable interactions We now examine the adaptation to a velocity-dependent field VF, an unstable position-dependent diverging field 373 A transverse movement [m/s] 0.8 1 [m/s] 0.4 0.5 0 0.5 1 time [s] 1.5 2 longitudinal movement hand speed 1.5 0 B longitudinal movement After adaptation, external force in the DF fields will be zero since the adapted movement is along {x = 0}. Furthermore, we assumed a stiffness increase Ki opposite to the environmental stiffness. In Fig. 7, it can be observed that the stiffness was elongated in the direction of instability for each of the three unstable fields. While this result is consistent with the result of Burdet et al. (2001) for DF{x=0} , no published data are available for the other destabilizing force fields yet. In the stable interaction with VF, the increase in stiffness was approximately along the direction of the external force, as was also found experimentally for movements (Franklin et al. 2003b) and static postures (Gomi and Osu 1998; Perreault et al. 2001). The time at which the stiffness ellipses were computed was 200 ms after movement onset. 4 Discussion 0.1 m 100 N/m C transverse movement Fig. 6a–c. Effect of movement speed on impedance. The hand speed during movement is shown in a. Stiffness is simulated for longitudinal (b) and transverse (c) movements with increasing peak velocities from light grey to medium grey (2×) to dark grey (4×). The longitudinal movements start at (0,0.3) m and end at (0,0.55) m. Transverse movements start at (−0.2, 0.45) m and end at (0.2,0.45) m. In contrast to Fig. 5, the external dynamics were not included in this simulation DF (Burdet et al. 2001; Franklin et al. 2003b), and two other similar DFs with different directions of destabilization. To facilitate comparison with published data, we considered a longitudinal movement from (0, 0.3) to (0, 0.55) m with a duration of 0.6 s. We considered the end state of adaptation where the mean trajectory over several trials would be similar to the planned trajectory. The force (in N) and impedance (in N/m) were computed using the following:     0 0 13 −18 (14) v, Ki = VF : FE = − 0 0 18 13     450 0 450 0 DF{x=0} : FE = − x, Ki = 0 0 0 0     315 0 315 0 DF{y=x} : FE = − x, Ki = 0 315 0 315     315 0 315 0 DF{y=−x} : FE = − x, Ki = . 0 −315 0 −315 The model introduced in this paper assumes that impedance depends principally on the force exerted by the muscles that span the joint and the environmental instability. This corresponds to reciprocal activation of agonist and antagonist muscles groups and their coactivation, recently postulated (Franklin et al. 2003c) to explain the control of unstable tasks. A comparison with published data demonstrated the predictive power of this model and in turn suggests that impedance of a limb exerting a force on the environment does not depend on whether the force is produced to move the arm or to interact with the environment. Using measurements of stiffness in static interactions with different levels of force applied in various directions, it becomes possible to predict stiffness during arbitrary movements adapted to a known stable or unstable dynamic environment. The simulations were restricted to two-dimensional horizontal arm movements, corresponding to published data. However, the model is valid for movements of the (possibly redundant) arm or leg in threedimensional space. This model was developed assuming that the musculoskeletal system has simple joints and uses a jointbased approach; it does not consider complex muscle mechanics and geometry. Therefore, we expect that this model may not be able to reproduce the adaptation to all environments equally well. In particular, the coupling of coactivation (i.e., stiffness) and reciprocal activation (i.e., force) is probably more complex than modeled here (Perreault et al. 2002). The stiffness dependence on torque magnitude was assumed to be linear and is expected to be valid up to at least 30% of the maximal voluntary contraction (Franklin and Milner 2003a). Many studies have focused on the adaptation that takes place when movements are repeated in novel dynamics (Shadmehr and Mussa-Ivaldi 1994; Shadmehr and Holcomb 1997; Conditt and Mussa-Ivaldi 1997; Burdet et al. 2001; Franklin et al. 2003c). The present model can only predict force and impedance on the first trial (before adaptation) or after learning. If the environmental interaction is unknown (e.g., a hazardous or inaccessible environment) or if learning transients are required (e.g., for 374 DF{y=x} VF DF{x=0} NF NF 100 N/m DF{y=-x} rehabilitation or human modeling), a learning algorithm predicting the evolution of force and impedance trial after trial would be necessary. It is likely that reflexes, muscle elasticity, motor noise, and feedforward commands will have to be considered to produce the correct interplay of dynamics and transients. In contrast to such a complex model of the adaptive controller of the human arm (Burdet et al. 2004), the very simple model introduced in this paper requires little computation. It can be used to simulate motion and easily integrated in the control of haptic interfaces. It may contribute to developing stable and optimal control for haptic interfaces and robots working with humans. Acknowledgements. 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