Estimation of Endpoint Impedance of a 2D Parallel Manipulator Using Numerical Simulation Experiments

Designing robots for physical interaction with humans pose unique design challenges. One such challenge is for the robots to be able to provide a wide range of mechanical impedance. In particular, for the design of an interactive robot for measuring human arm impedance during overground tasks, it is crucial for the robot output impedance to be low. Even though, with the help of control algorithms, it is possible to achieve a high endpoint impedance characteristic in a manipulator, it may not be feasible to go below certain threshold limit depending upon the inherent impedance characteristics such as inertia and damping. Thus, it is beneficial to estimate these parameters as early as in design phase and modify the design according to the end goal on the endpoint impedance.

Following our previous work in which we have suggested a method to estimate the endpoint impedance, in this work we present a simulation method for a five-link parallel manipulator for estimating the effective inertia (EI) and effective damping (ED) at the endpoint. First, the Euler-Lagrangian Method was used to construct the dynamic equation for robot in actuator space (motor shaft angle, θ), such that M(θ) d²θ/dt²+ V_m(θ,dθ/dt) dθ/dt+ tau_r(dθ/dt)= tau+ J^T(θ) F, where M, V_m, J, tau_r, tau and F are matrix/vector for inertia, coriolis/centripetal effect, Jacobian, friction, actuator torque and external force respectively. Eventually, this should be equivalent to the Cartesian form with EI and ED, such that EI d²x/dt²+ ED dx/dt= F, where x represents endpoint displacement in 2D. EI, which is the function of θ only, can be derived using M and J, such that EI=J^-TMJ^-1. On the other hand, ED is not only affected by joint friction but also is a function of motion parameters (such as the shaft velocity and position), which introduce complex non-linear effects, and hence can only be estimated numerically. Our method of estimating ED is inspired by the experimental method for estimating human arm impedance. In our simulation method, we choose a specific endpoint position inside the workspace and apply an external force in ‘n’ different directions (equally spaced but starting at 5° offset CW from the axis of symmetry of the robot). Then, the resulting motion parameters for each direction is recorded after 0.05 seconds into the dynamics simulation to prevent the configuration to change significantly. By linearly relating the endpoint movement to the applied forces, ‘n’ number of 2x2 damping matrices cab be found. Lastly, the EI and ED matrix for that specific endpoint position is estimated using linear regression of these ‘n’ matrices. The procedure is repeated for other endpoint positions of interest. To validate if this method works correctly, we assumed a case where ED=0, and compared the numerically found EI with the theoretically found J^-TMJ^-1 at each endpoint positions. We found that the two values are very similar as long as the provided external force is a step function. Hence, it is reasonable to claim that ED found using the same method is an accurate estimation of the damping. With the EI and ED matrices identified, the norms of these matrices is used to compute a single scalar magnitude of inertia [Kg] or damping [Ns/m]. Between the six configurations of θ, the average inertia was found to be 0.3442 Kg with a maximum reaching to 0.408 Kg. Likewise, the average damping coefficient was found to be 1.1783 Ns/m with maximum damping coefficient of 1.6601 Ns/m. This inherent endpoint impedance behavior is acceptable for human-robot interaction.

The proposed method to estimate the endpoint impedance of a five-bar robot arm can help design an interactive robot with specific inherent endpoint impedances as early as in the design phase. We expect our work to especially benefit the design of interactive robots for human-robot collaboration, cooperation and/or assistance.