Hand Dynamics and Control

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Chapter 6 - Hand Dynamics and Control

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In this chapter, we study the dynamics and control of a set of robots performing a coordinated task. Our primary example will be that of a multifingered robot hand manipulating an object, but the formalism is considerably broader. It allows a unified treatment of dynamics and control of robot systems subject to a set of velocity constraints, generalizing the treatment given in Chapter 4.

Chapter Summary

The following are the key concepts covered in this chapter:

1. The dynamics of a mechanical system with Lagrangian , subject to a set of Pfaffian constraints of the form
   

   can be written as

   

   where is the vector of Lagrange multipliers. The values of the Lagrange multipliers are given by

   
2. The Lagrange-d'Alembert formulation of the dynamics represents the motion of the system by projecting the equations of motion onto the subspace of allowable motions. If and the constraints have the form
   

   then the equations of motion can be written as

   

   In the special case that the constraint is integrable, these equations agree with those obtained by substituting the constraint into the Lagrangian and then using the unconstrained version of Lagrange's equations.

3. The dynamics for a multifingered robot hand with joint variables and (local) object variables , subject to the grasp constraint
   

   is given by

   

   where and

   

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

4. 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 span the null space of , and represents the internal motions of the system. For nonmanipulable systems, we choose a matrix which spans the space of allowable object trajectories and write the constraints as

   

   where 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.

5. The kinematics of tendon-driven systems are described in terms of a set of extension functions, , which measures the displacement of the tendon as a function of the joint angles of the system. If a vector of tendon forces is applied at the end of the tendons, the resulting joint torques are given by
   

   where is the coupling matrix:

   

   A tendon-system is said to be force-closure at a point if for every vector of joint torques, , there exists a set of tendon forces which will generate those torques.

6. The equations of motion for a constrained robot system are described in terms of the quantities , , and . When correctly defined, the quantities satisfy the following properties:
   • is symmetric and positive definite.
   • is a skew-symmetric matrix.
   \end{enumerate} 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 is the generalized force in object coordinates (determined by the control law) and 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.

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