Cartesian Impedance Control for Physical Human–Robot Interaction Using Virtual Decomposition Control Approach

Abstract

This paper presents and verifies a novel Cartesian impedance control of the flexible joint manipulator for physical human–robot interaction application based on virtual decomposition control approach. Firstly, the Cartesian impedance control based on virtual decomposition control (VDC) is presented, and the asymptotical stability of the controller is proven by Lyapunov stability theorem. Compared with traditional methods based on singular perturbation, this method can greatly reduce the computational loads and is more suitable for real-time application. Then, a Cartesian force-feedback path planning combined with Cartesian impedance control based on VDC was used to keep the real contact force within the desired value to protect the manipulator and objects as a force, position, velocity and acceleration sensor, and the sensor can be configured freely by regulating the stiffness, damping and inertia, so the manipulator can interact with human (or unknown environment) in a friendly manner. The experimental results illustrate the validity of the developed VDC-based impedance control approach.

Appendix

1. How the joint torques are measured by the strain gauges, as it is mentioned in lines 54 and 55 on page 9.

When the beam is subjected to torque deformation, the strain gauge will produce shear strain, the resistance value of strain gauge changes. The four strain gauges on the beam constitute a full-bridge circuit, the bridge output voltage through the amplifier into the A/D conversion device, then into the computer. The strain signal is calibrated by least squares method and converted to the corresponding torque.

The friction, stiffness and damping parameters of the robot were identified as the following.

• 1. The friction parameters identification

The friction model from the LuGre steady-state friction, payload-dependent friction and motor-position-based friction is expressed as:

$$\tau_{\text{F}} = g_{\tau } (\tau )(\alpha_{0} + \alpha_{1} e^{{ - (\theta /v_{s} )^{2} }} )\text{sgn} (\dot{\theta }) + \alpha_{2} \dot{\theta } + H(\theta ) = Y(\tau ,\dot{\theta })K_{\text{F}}$$

$$g_{\tau } (\tau ) = (1 + g_{1} \left| \tau \right| + g_{2} \left| \tau \right|^{2} )$$

It covers Stribeck velocity $v_{\text{s}}$, static friction at zero payload ($\alpha_{0} + \alpha_{1}$), viscous friction $\alpha_{2}$ and position-based friction $H(\theta )$. Additionally, with $g_{1} > 0$ and $g_{2} > 0$, $g_{\tau } (\tau )$ is used to emulate the load-dependent static friction effects. The complete friction model is characterized by four uncertain parameter vectors:

$$K_{F} = \left[ {\begin{array}{*{20}c} {\alpha_{0} } & {\alpha_{1} } & {\alpha_{2} } & {H(\theta )} \\ \end{array} } \right]^{T} \in R^{4}$$

Through the experiment, we obtain the joint friction–velocity curve and friction–motor angle curve at static velocity as the following:

• 2. The stiffness and damping parameters identification

Under joint impedance control, when the joint is in contact with the mechanical limit. According to the equation τ = K(θ − q), q value is constant, τ can be measured by joint torque sensor, and θ can be measured by Joint position sensor, and we can easily calculate the stiffness coefficient K.

Joint damping in parallel to the joint spring, the D can be calculated as the following:

$$B\ddot{\theta } + K(\theta - q) + D(\dot{\theta } - \dot{q}) = \tau$$

where B is the motor inertias. $K(\theta - q) = \tau$, so $B\ddot{\theta } = D(\dot{q} - \dot{\theta })$.