# [MLS94] Hand Dynamics and Control

Unified treatment of dynamics and control of robot systems subject to a set of velocity constraints.

## Robot Hand Dynamics

Object dynamics are given by the Newton-Euler equations:

The dynamics for a multifingered robot hand with joint variables $\thetab \in \mathbb{R}^n$ and local object variables $\xb \in \mathbb{R}^p$, subject to the grasp constraints

is given by

where $\qb = (\thetab, \xb)$ and

These same equations can be applied to a large number of other robotic systems by choosing $G$ and $J_h$ appropriately.

## Redundant and Nonmanipulable Robot Systems

For *redundant* and/or *nonmanipulable* robot systems, the hand Jacobian is not invertible, resulting in a more complicated derivation of the equations of motion. For redundant systems, the constraints can be extended to the form

where the rows of $K_h$ span the null space of $J_h$, and $\vb_N$ represents the internal motions of the system. For nonmanipulable systems, we choose a matrix $H$ which spans the space of allowable object trajectories and write the constraints as

where $\xbdot = H(\qb) \omegab$ represents the object velocity. In both the redundant and nonmanipulable cases, the augmented form of the constraints can be used to derive the equations of motion and put them into the standard form given above.

## Control of Robot Hand

The equations of motion for a constrained robot system are described in terms of the quantities $\tilde{M}(\qb), \tilde{C}(\qb,\qbdot)$ and $\tilde{N}(\qb, \qbdot)$. When correctly defined, the quantities satisfy the following properties:

- $\tilde{M}(\qb)$ is symmetric and positive definite.
- $\dot{\tilde{M}}(\qb) - 2\tilde{C}$ is a skew-symmetric matrix.

Using these properties it is possible to extend the controllers presented in Chapter 4 to the more general class of systems considered in this chapter. For a multifingered hand, an extended control law has the general form

where $F$ is the generalized force in object coordinates (determined by the control law) and $f_N$ is an internal force. The internal forces must be chosen so as to insure that all contact forces remain inside the appropriate friction cone so that the fingers satisfy the fundamental grasp constraint at all times.