Collision avoidance analysis of human–robot physical interaction based on null-space impedance control of a dynamic reference arm plane

Abstract

When the terminal upper limb rehabilitation robot is used for motion-assisted training, collisions between the manipulator links and the human upper limb may occur due to the null-space self-motion of the redundant manipulator. A null-space impedance control method based on a dynamic reference arm plane is proposed to realize collision avoidance during human–robot physical interaction motion for the collision problem between the manipulator links and the human upper limb. Firstly, a dynamic model and a Cartesian impedance controller of the manipulator are established. Then, the null-space impedance controller of the redundant manipulator is established based on the dynamic reference plane, which manages the null-space self-motion of the redundant manipulator to prevent collision between the manipulator links and the human upper limb. Finally, it is experimentally verified that the method proposed in this paper can effectively manage the null-space self-motion of the redundant manipulator, and thus achieve collision avoidance during the human–robot physical interaction motion. This research has significant potential in improving the safety and feasibility of motion-assisted training with rehabilitation robots.

Graphical abstract

Appendix A

Theorem 1: Suppose the mathematical model of a system is $\dot{x}=f(x)$, $x\in {\mathrm{R}}^{\mathrm{n}}$, and the equilibrium point of the system is ${x}_{\mathrm{e}}$. Let $\mathrm{S}(x)$ be a ${C}^{1}$ semi-positive definite function and the derivative for time be semi-negative definite. As shown in Eq. (1A),

$$\dot{\mathrm{S}}(x)=\frac{\partial \mathrm{V}\left(x\right)}{\partial x}f(x)\le 0.$$

(1A)

Suppose $\Omega$ be the set of maximal positive invariants contained in $\{x\in {\mathrm{R}}^{\mathrm{n}}|\mathrm{V}(x)=0\}$. If ${x}_{\mathrm{e}}$ is conditionally asymptotically stable for $\Omega$, so this is a stable equilibrium for $\dot{x}=f(x)$.

Proof: This theorem was proposed by Iggidr et al. [37] The detailed proof procedure is given in the literature [38].

Theorem 2: Assume that the mathematical model of the system is

$$\left\{\begin{array}{c}\dot{x}={\mathrm{g}}_{1}(x)+{\mathrm{g}}_{2}(x)u\\ =h(x)\end{array}\right..$$

(1B)

where the state quantity $x\in {\mathrm{R}}^{\mathrm{n}}$, the input quantity $u\in {\mathrm{R}}^{\mathrm{m}}$ and the output quantity $y\in {\mathrm{R}}^{\mathrm{m}}$ are strictly passive outputs of the output quantity $y=\mathrm{h}(x)$. Further, let $\Omega$ be the largest positive invariant set contained in $\{x\in {\mathrm{R}}^{\mathrm{n}}|\mathrm{h}(x)=0\}$. If ${x}_{\mathrm{e}}$ is conditionally asymptotically stable for $\Omega$, so this is a stable equilibrium for $u=0$.

Proof: This theorem was proposed by Schaft V D and Arjan, and the detailed proof is given in the literature [39].

When the system is asymptotically stable, it needs to satisfy the condition ${\ddot{\mathbf{x}}}_{\mathrm{e}}=0$, ${\ddot{\mathbf{x}}}_{\mathrm{n},\mathrm{e}}=0$, ${\dot{\mathbf{x}}}_{\mathrm{e}}=0$, ${\dot{\mathbf{x}}}_{\mathrm{n},\mathrm{e}}=0$, $\mathbf{x}={\mathbf{x}}_{\mathrm{e}}$ and ${\mathbf{x}}_{\mathrm{n}}={\mathbf{x}}_{\mathrm{n},\mathrm{e}}$, and there is no external contact at the point of asymptotic equilibrium stability, i.e., ${\mathbf{F}}_{\mathrm{ext},\mathrm{c}}=0$ and ${\mathbf{F}}_{\mathrm{ext},\mathrm{n}}=0$. Furthermore, let the mass matrices of the Cartesian impedance controller and the null-space impedance controller be ${{\varvec{\Lambda}}}_{\mathrm{c}}={\mathbf{M}}_{\mathrm{c}}(\mathbf{q})$ and ${{\varvec{\Lambda}}}_{\mathrm{n}}=\mathbf{N}(\mathbf{q})\mathbf{M}(\mathbf{q})\mathbf{N}{(\mathbf{q})}^{\mathrm{T}}$, respectively, and introduce ${\mathbf{C}}_{\mathrm{c}}(\mathbf{q},\dot{\mathbf{q}})$ and ${{\varvec{\mu}}}_{\mathrm{n}}={{\varvec{\Lambda}}}_{\mathrm{n}}\mathbf{N}(\mathbf{q})(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\mathbf{N}{(\mathbf{q})}^{\mathrm{T}}+\mathbf{M}(\mathbf{q})\dot{\mathbf{N}}{(\mathbf{q})}^{\mathrm{T}})$. In turn, the closed-loop control system Eq. (6) and Eq. (17) are transformed into Eq. (1C) and Eq. (1D).

$${{\varvec{\Lambda}}}_{\mathrm{c}}\ddot{\mathbf{x}}+({\mathbf{C}}_{\mathrm{c}}(\mathbf{q},\dot{\mathbf{q}})+{\mathbf{D}}_{\mathrm{c}})\dot{\mathbf{x}}+{\mathbf{K}}_{\mathrm{c}}(\mathbf{x}-{\mathbf{x}}_{\mathrm{e}})\text{=}{\mathbf{F}}_{\mathrm{ext},\mathrm{c}}$$

(1C)

$${{\varvec{\Lambda}}}_{\mathrm{n}}{\ddot{\mathbf{x}}}_{\mathrm{n}}+({{\varvec{\mu}}}_{\mathrm{n}}+{\mathbf{D}}_{\mathrm{n}}){\dot{\mathbf{x}}}_{\mathrm{n}}+{\mathbf{K}}_{\mathrm{n}}({\mathbf{x}}_{\mathrm{n}}-{\mathbf{x}}_{\mathrm{n},\mathrm{e}})\text{=}{\mathbf{F}}_{\mathrm{ext},\mathrm{n}}$$

(1D)

The stability analysis of the closed-loop control system formulas (1C) and (1D) is proved by Theorem 2, which leads to the construction of a semi-positive definite Lyapunov function. As shown in Eq. (1F).

$$\mathrm{V}(\mathbf{x},\dot{\mathbf{x}})=\frac{1}{2}{\dot{\mathbf{x}}}^{\mathrm{T}}{{\varvec{\Lambda}}}_{\mathrm{c}}\dot{\mathbf{x}}+\frac{1}{2}{(\mathbf{x}-{\mathbf{x}}_{\mathrm{e}})}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{c}}(\mathbf{x}-{\mathbf{x}}_{\mathrm{e}})$$

(1F)

Take the time derivative of Eq. (1F), and ${\dot{\mathbf{M}}}_{\mathrm{c}}(\mathbf{q})-{\mathbf{C}}_{\mathrm{c}}(\mathbf{q},\dot{\mathbf{q}})$ is the antisymmetric matrix, and combine Eq. (1C) to obtain Eq. (1G).

$$\dot{\mathrm{V}}(\mathbf{x},\dot{\mathbf{x}})=-{\dot{\mathbf{x}}}^{\mathrm{T}}{\mathbf{C}}_{\mathrm{c}}\dot{\mathbf{x}}-{\dot{\mathbf{x}}}^{\mathrm{T}}{\mathbf{F}}_{\mathrm{ext},\mathrm{c}}$$

(1G)

It is obvious from Eq. (1C) that when ${\mathbf{F}}_{\mathrm{ext},\mathrm{c}}=0$, $\dot{\mathrm{V}}(\mathbf{x},\dot{\mathbf{x}})\le 0$ satisfies the asymptotic equilibrium stability condition of Theorem 1. According to Theorem 2, we must prove that the system is conditionally asymptotically stable for the largest positive invariant set in $\{(q,\dot{x},{\dot{x}}_{\mathrm{n}})\in {\mathrm{R}}^{2\mathrm{n}}|\dot{x}=0\}$. Consider further the case of free motion, i.e., ${\mathbf{F}}_{\mathrm{ext},\mathrm{c}}=0$ and ${\mathbf{F}}_{\mathrm{ext},\mathrm{n}}=0$. The maximum positive invariant set maintained by $\dot{\mathbf{x}}=0$ can be derived from Eq. (1C) as Eq. (1H).

$$\mathrm{U}=\{(\mathbf{x},\dot{\mathbf{x}},{\dot{\mathbf{x}}}_{\mathrm{n}})|\mathbf{x}={\mathbf{x}}_{\mathrm{e}},\dot{\mathbf{x}}=0\}$$

(1H)

Furthermore, since U is a positive invariant set, all trajectories starting from U stay in U and thus the Lyapunov function is constructed as Eq. (1I).

$${\mathrm{W}}_{\mathrm{U}}({\mathbf{x}}_{\mathrm{n}},{\dot{\mathbf{x}}}_{\mathrm{n}})=\frac{1}{2}{\dot{\mathbf{x}}}_{\mathrm{n}}^{\mathrm{T}}{{\varvec{\Lambda}}}_{\mathrm{n}}{\dot{\mathbf{x}}}_{\mathrm{n}}+\frac{1}{2}{({\mathbf{x}}_{\mathrm{n}}-{\mathbf{x}}_{\mathrm{n},\mathrm{e}})}^{\mathrm{T}}{\mathbf{K}}_{\mathrm{n}}({\mathbf{x}}_{\mathrm{n}}-{\mathbf{x}}_{\mathrm{n},\mathrm{e}})$$

(1I)

The time derivative of Eq. (1I) and the combination of Eq. (1C) and Eq. (1D) yields Eq. (1J).

$${\dot{\mathrm{W}}}_{\mathrm{U}}({\mathbf{x}}_{\mathrm{n}},{\dot{\mathbf{x}}}_{\mathrm{n}})=-{\dot{\mathbf{x}}}_{\mathrm{n}}^{\mathrm{T}}{\mathbf{D}}_{\mathrm{c}}{\dot{\mathbf{x}}}_{\mathrm{n}}-{\dot{\mathbf{x}}}_{\mathrm{n}}^{\mathrm{T}}{\mathbf{F}}_{\mathrm{ext},\mathrm{n}}$$

(1J)

According to Eq. (1D), when ${\mathbf{F}}_{\mathrm{ext},\mathrm{c}}=0$, ${\dot{\mathrm{W}}}_{\mathrm{U}}({\mathbf{x}}_{\mathrm{n}},{\dot{\mathbf{x}}}_{\mathrm{n}})=-{\dot{\mathbf{x}}}_{\mathrm{n}}^{\mathrm{T}}{\mathbf{D}}_{\mathrm{c}}{\dot{\mathbf{x}}}_{\mathrm{n}}\le 0$ satisfies the asymptotic equilibrium stability condition of theorem 1. And the conditional stability of U can be obtained from ${\dot{\mathrm{W}}}_{\mathrm{U}}({\mathbf{x}}_{\mathrm{n}},{\dot{\mathbf{x}}}_{\mathrm{n}})=-{\dot{\mathbf{x}}}_{\mathrm{n}}^{\mathrm{T}}{\mathbf{D}}_{\mathrm{c}}{\dot{\mathbf{x}}}_{\mathrm{n}}\le 0$. To prove the asymptotic stability of U under the condition, we can follow LaSalle's invariance principle. It follows that all trajectories in U converge to the largest contained positive invariant set Eq. (1K).

$$\{(\mathbf{x},\dot{\mathbf{x}},{\dot{\mathbf{x}}}_{\mathrm{n}})|\mathbf{x}={\mathbf{x}}_{\mathrm{e}},\dot{\mathbf{x}}=0,{\dot{\mathbf{x}}}_{\mathrm{n}}=0\}$$

(1K)

In summary, based on the superposition of the Cartesian impedance controller and the null-space impedance controller, the whole closed-loop control system satisfies the asymptotic stability.