Changes in Constraint

There are only two types of changes in constraint: 1) changes from a manifold of lesser constraint (higher dimension) to one of more constraint (lower dimension), or 2) changes from lower dimension to higher dimensional surfaces. It is impossible to cross between two manifolds of the same dimension without rst passing through either the larger surface they are embedded in or through their intersection.

An impact wrench will result from the rst kind of change if the velocity causing the transition is not tangent to the new more constrained surface. Impacts result from transition velocities which violate the constraints of the new surface. For example, to avoid an impact wrench a point in ^<3 which is going to contact a plane must have a crossing velocity which lies entirely in the plane. This sort of motion can be seen as a limiting condition on a smooth transition motion. This can occur when two constraint manifolds smoothly blend into each other.

The second type of transition may or may not result in an apparent impact wrench.

The same ideal point will not experience an impact when it leaves the plane, because the higher dimensional space can accommodate the entrance velocity. However, any real robot has some built up potential spring energy in the sensor structure, the contacting linkage, and the robot joints. During contact, this energy is being contained by the constraint surface. If contact is lost abruptly, a contact loss impact occurs because of the sudden release of this energy. Abruptness is a function of the bandwidth and damping of the sensor and the speed of the transition.

For the rst type of transition we need to distinguish between stable and unstable transitions. An example of an unstable transition is an object being pressed into a surface and then dragged along that surface. When the orientation of the surface discontinuously changes (i.e. we cross an edge), a sudden change in the orientation causes the object to lose contact with the surface and a loss impact results.

A transition is stable if the transition into the intersection between two manifolds remains in the intersection instead of continuing onto the next manifold. To formalize this let ^Mi and ^Mj be two constraints in the conguration space with non-empty

4.3: Dynamics

71

intersection ^Mi;j = ^Mi^TMj. Let their co-tangent bundles be dened so that the set of wrenches they can support is given by the positive convex combination over the basis vectors in the co-tangent space and the disturbance vectors. The set of wrenches supported on ^Mi at

x

is then Supi(

x

) = Convex(^Ti(x);Range[

w

d](x)).

The range value models the friction component which also depends upon the applied wrench.

A state

x

and control

v

d will be totally stable, resulting in no motion, if the applied control can be entirely supported by the constraint

K

r

x

~^;

B

r

v

d ²Sup_i(

x

): (4:45) A less strict form of stability is that the control plus the resulting constraints result in a motion in the tangent space of the intersection manifold ^Mi;j. This denition is more useful because it states that motions on a manifold stay on a manifold. This will be the case if at state

x

and control (

x

d;

v

d) there is a nonnegative solution for

in

(

C

x^T

B

^;r¹

C

x)=

C

x^T

B

^;r¹(^;

K

r

x

~+

B

r

v

d+ E[

w

d]): (4:46) Such a triple (

x

;

x

d;

v

d) will be called manifold stable.

In order for a transition from ^Mi to ^Mj to remain on the intersection, several conditions must hold. First, the control

v

d and the transition state

x

must be manifold stable on ^Mi;j. Second, the control and state must be only manifold stable on ^Mi. If this condition does not hold, the robot will become stuck on ^Mi and will never transition. Lastly, the velocity that results from the control on^Mi must have a component that causes motion toward the intersection.

In an ideal stable impact the momentum changes discontinuously to bring the new velocity into the new tangent space. Let the initial manifold be ^Mi and the new manifold be ^Mj. Let

v

new be the velocity after transition and the velocity before transition be

v

trans.

v

newmust lie in the tangent space of^Mi;jso that

C

i;j(

x

)^T

v

new = 0. The change in momentum is caused by an impulse which lies entirely in the cotangent space of ^Mj. The equation for the change in momentum is

H

t(

x

)

v

new =

H

t(

x

)

v

trans+

C

j(

x

)

k

: (4:47) Using this relationship, the constraint gives

k

= ^;(

C

_Tj

H

t

C

j)^;1

C

_Tj

H

^;_t¹

p

old (4.48)

w

impulse =

C

j

k

(4.49)

p

new = (

Id

^;

C

j(

C

Tj

H

t

C

j)^;1

C

Tj

H

^;t¹)

p

old (4.50)

72

Chapter 4: Manipulation and Constraints

Mi

Mj

Mi,j v_trans

v_new

Figure 4.9: Impact transition from manifold ^Mi to manifold^Mi;j.

for the impulse force and new momentum. The impulse force is generated over a small window of time

w

impulse =^Z_t^t+tE[

w

(t)]dt: (4:51)

The impact wrench adds into the measured wrench. The idealized model for E[

w

] is a delta function at time t. However, in actuality the workpiece bounces several times, in a complex way, before settling onto the new constraint. Energy is lost on every bounce due to damping in the robot and at the interface. The integral can be computed from the data if the beginning and end of the impact can be identied.

Chapter 7 discusses this identication problem. The change in momentum can be used to help determine the new constraint manifold.

For an unstable impact, there is a release in potential energy. The change in the average force stored in the spring is

K

(E[~

x

]after^;E[~

x

]before) = E[

C

xbefore]^;E[

C

xafter]: (4:52) This dierence in wrench adds an impulse wrench into the measured wrench. The impulse causes a transient response which slowly dies out. The direction of change can be found by identifying the peak wrench.

4.4: Conclusion

73

4.4 Conclusion

This chapter formulated the constraints and dynamics of rigid bodies in the cong-uration space of allowed motions. The congcong-uration space was shown to have an intrinsic topology graph. Each node in the graph represented a region of constant dynamics. The dynamics were shown to depend critically on the type and geomet-ric description of the constraint. It was shown that dierent geometgeomet-ric descriptions could result in the same constraint.

For constant velocity motions on a single conguration space manifold, the stationary solution to the expected value of the motion was shown to take the form of a point sliding on the manifold pulled by a spring and damper. The statistics of the motion and the measured wrench were shown to depend upon not only the control and contact disturbances, but also on the direction and magnitude of the motion.

Transitions between nodes in the graph were shown to correspond to transitory events. Transitions to manifolds with more constraint were shown to be either stable or unstable. Stable transitions almost always result in impacts. Unstable transitions result in loss impacts if there is stored elastic energy.

74

Chapter 4: Manipulation and Constraints

Dans le document Contact Sensing: A Sequential Decision Approach to Sensing Manipulation Contact (Page 70-75)