Hand Dynamics and Control - Revision history

Murray: /* Chapter Summary */

2010-03-28T20:14:40Z

Chapter Summary

Murray: /* Chapter Summary */

2010-03-28T18:35:08Z

Chapter Summary

Murray: /* Chapter Summary */

2010-03-28T18:34:49Z

Chapter Summary

Murray: /* Chapter Summary */

2009-07-25T02:49:03Z

Chapter Summary

Murray at 02:42, 25 July 2009

2009-07-25T02:42:47Z

Murray at 02:07, 22 July 2009

2009-07-22T02:07:45Z

Murray at 02:05, 22 July 2009

2009-07-22T02:05:46Z

Murray at 02:05, 22 July 2009

2009-07-22T02:05:22Z

Murray at 02:01, 22 July 2009

2009-07-22T02:01:28Z

New page

{{chapter header|Multifingered Hand Kinematics|Hand Dynamics and Control|Nonholonomic Behavior in Robotic Systems}}

@@ Line 14: / Line 14: @@
 <li> The dynamics of a mechanical system with Lagrangian <amsmath>L(q, \dot
 q)</amsmath>, subject to a set of ''Pfaffian constraints'' of the form
-<center><amsmath>
+<blockquote><center><amsmath>
    A(q) \dot q = 0 \qquad A(q) \in {\mathbb R}^{k \times n},
-</amsmath></center>
+</amsmath></center></blockquote>
 can be written as
 <center><amsmath>

@@ Line 36: / Line 36: @@
 {\mathbb R}^{(n-k) \times k}</amsmath> and the constraints have the form
 <center><amsmath>
-   \dot q_2 = {\cal A(q)} \dot q_1,
+   \dot q_2 = {\cal A}(q) \dot q_1,
 </amsmath></center>
 then the equations of motion can be written as

@@ Line 36: / Line 36: @@
 {\mathbb R}^{(n-k) \times k}</amsmath> and the constraints have the form
 <center><amsmath>
-   \dot q_2 = \cal A(q) \dot q_1,
+   \dot q_2 = {\cal A(q)} \dot q_1,
 </amsmath></center>
 then the equations of motion can be written as

← Older revision		Revision as of 02:49, 25 July 2009
Line 9:		Line 9:

	== Chapter Summary ==		== Chapter Summary ==
		+
		+	The following are the key concepts covered in this chapter:
		+	<ol>
		+	<li> The dynamics of a mechanical system with Lagrangian <amsmath>L(q, \dot
		+	q)</amsmath>, subject to a set of ''Pfaffian constraints'' of the form
		+	<center><amsmath>
		+	A(q) \dot q = 0 \qquad A(q) \in {\mathbb R}^{k \times n},
		+	</amsmath></center>
		+	can be written as
		+	<center><amsmath>
		+	\frac{d}{dt} \frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} + A^T(q) \lambda - \Upsilon = 0,
		+	</amsmath></center>
		+	where <amsmath>\lambda \in {\mathbb R}^k</amsmath> is the vector of ''Lagrange multipliers''.
		+	The values of the Lagrange multipliers are given by
		+	<center><amsmath>
		+	\lambda =
		+	(A M^{-1} A^T)^{-1} \left(
		+	A M^{-1}(F - C\dot q - N) + \dot A \dot q
		+	\right).
		+	</amsmath></center>
		+	</li>
		+
		+	<li> 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 <amsmath>q = (q_1, q_2) \in
		+	{\mathbb R}^{(n-k) \times k}</amsmath> and the constraints have the form
		+	<center><amsmath>
		+	\dot q_2 = \cal A(q) \dot q_1,
		+	</amsmath></center>
		+	then the equations of motion can be written as
		+	<center><amsmath>
		+	\left(\frac{d}{dt} \frac{\partial L}{\partial \dot q_1} - \frac{\partial L}{\partial q_1} - \Upsilon_1\right) +
		+	{\cal A}^T \left(\frac{d}{dt} \frac{\partial L}{\partial \dot q_1} - \frac{\partial L}{\partial q_2} -
		+	\Upsilon_2\right)
		+	= 0.
		+	</amsmath></center>
		+	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.
		+	</li>
		+
		+	<li> The dynamics for a ''multifingered robot hand'' with joint variables
		+	<amsmath>\theta \in {\mathbb R}^n</amsmath> and (local) object variables <amsmath>x \in {\mathbb R}^p</amsmath>,
		+	subject to the grasp constraint
		+	<center><amsmath>
		+	J_h(\theta, x) \dot \theta = G^T(\theta, x) \dot x,
		+	</amsmath></center>
		+	is given by
		+	<center><amsmath>
		+	\tilde M(q) \ddot x + \tilde C(q, \dot q) \dot x + \tilde N(q, \dot
		+	q) = F,
		+	</amsmath></center>
		+	where <amsmath>q = (\theta, x)</amsmath> and
		+	<center><amsmath>
		+	\aligned
		+	\tilde{M} &= M_o + G J_h^{-T} M_f J_h^{-1} G^T \\
		+	\tilde{C} &= C_o + G J_h^{-T} \left(C_f J_h^{-1} G^T +
		+	M_f \frac{d}{dt} \left(J_h^{-1} G^T \right)\right) \\
		+	\tilde{N} &= N_o + G J_h^{-T} N_f \\
		+	F &= G J_h^{-T} \tau.
		+	\endaligned
		+	</amsmath></center>
		+	These same equations can be applied to a large number of other robotic
		+	systems by choosing <amsmath>G</amsmath> and <amsmath>J_h</amsmath> appropriately.
		+	</li>
		+
		+	<li> 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
		+	<center><amsmath>
		+	\underbrace{\begin{bmatrix} J_h \\ K_h \end{bmatrix}}_{\bar J_h} \dot\theta =
		+	\underbrace{\begin{bmatrix} G^T & 0 \\ 0 & I \end{bmatrix}}_{\bar G^T}
		+	\begin{bmatrix} \dot x \\ v_N \end{bmatrix},
		+	</amsmath></center>
		+	where the rows of <amsmath>K_h</amsmath> span the null space
		+	of <amsmath>J_h</amsmath>, and <amsmath>v_N</amsmath> represents the ''internal motions'' of
		+	the system. For nonmanipulable systems, we choose a matrix <amsmath>H</amsmath> which
		+	spans the space of allowable object trajectories and write the
		+	constraints as
		+	<center><amsmath>
		+	J_h \dot\theta = \underbrace{G^T H}_{\bar G^T} w,
		+	</amsmath></center>
		+	where <amsmath>\dot x = H(q) w</amsmath> 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.
		+	</li>
		+
		+	<li> The kinematics of ''tendon-driven systems'' are described in terms
		+	of a set of ''extension functions'', <amsmath>h_i:Q \to {\mathbb R}</amsmath>, which measures
		+	the displacement of the tendon as a function of the joint angles of
		+	the system. If a vector of tendon forces <amsmath>f \in {\mathbb R}^k</amsmath> is applied
		+	at the end of the tendons, the resulting joint torques are given by
		+	<center><amsmath>
		+	\tau = P(\theta) f,
		+	</amsmath></center>
		+	where <amsmath>P(\theta) \in {\mathbb R}^{n \times p}</amsmath> is the ''coupling matrix'':
		+	<center><amsmath>
		+	P(\theta) = \frac{\partial h}{\partial \theta}^T(\theta).
		+	</amsmath></center>
		+	A tendon-system is said to be ''force-closure'' at a point <amsmath>\theta</amsmath>
		+	if for every vector of joint torques, <amsmath>\tau</amsmath>, there exists a set of
		+	tendon forces which will generate those torques.
		+	</li>
		+
		+	<li> The equations of motion for a constrained robot system are
		+	described in terms of the quantities <amsmath>\tilde M(q)</amsmath>, <amsmath>\tilde C(q, \dot
		+	q)</amsmath>, and <amsmath>\tilde N(q, \dot q)</amsmath>. When correctly defined, the
		+	quantities satisfy the following properties:
		+	* <amsmath>\tilde M(q)</amsmath> is symmetric and positive definite.
		+	* <amsmath>\Dot{\tilde M}(q) - 2 \tilde C</amsmath> 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
		+	<center><amsmath>
		+	\tau = J^T G^+ F + J^T f_N,
		+	</amsmath></center>
		+	where <amsmath>F</amsmath> is the generalized force in object coordinates (determined by
		+	the control law) and <amsmath>f_N</amsmath> 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.
		+	</li>
		+	</ol>

	== Additional Information ==		== Additional Information ==

← Older revision		Revision as of 02:42, 25 July 2009
Line 1:		Line 1:
	{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior}}		{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior}}
		+
		+	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 ==
		+
		+	== Additional Information ==

← Older revision		Revision as of 02:07, 22 July 2009
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−	{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior ~~in Robotic Systems~~}}	+	{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior}}

← Older revision		Revision as of 02:05, 22 July 2009
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−	{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior}}	+	{{chapter header\|Multifingered Hand Kinematics\|Hand Dynamics and Control\|Nonholonomic Behavior in Robotic Systems}}