An Introduction to Mobility of Cooperating Robots with Unactuated Joints and Closed-Chain Mechanisms

Transcription

1 An Introduction to Mobility of Cooperating Robots with Unactuated Joints and Closed-Chain Mechanisms Daniele Genovesi Interdepartmental Research Center "E.Piaggio" Faculty of Automation Engineering University of Pisa 23/05/2011

2 INDEX References 1

3 References Referenced articles [1] Antonio Bicchi. On the problem of decomposing grasp and manipulation forces in multiple whole limb manipulation. In: Int. Journal of Robotics and Autonomous Systems vol. 13 (1994), pp Elsevier Science, Oxford, UK. 3 (1994). [2] Antonio Bicchi and Domenico Prattichizzo. Manipulability of Cooperating Robots with Unactuated Joints and Closed-Chain Mechanisms. In: IEEE Transactions on Robotics and Automation Download link Results proposed in the above articles apply directly to the analysis of cooperating robots, parallel robots, dextrous robotic hands and legged vehicles, and, in general, to closed kinematic chains.

4 System denition References MANIPULATION The approach we follow to analyze kinematics and statics of closed-chain mechanical system is to consider them as embodiments of a cooperative manipulation paradigm, where multiple robotic limbs (or ngers) interact with an object at a number of contacts. The object is the reference member of the mechanism, whose motions and forces are the ultimate goal of analysis. Contacts represent in fact unactuated kinematic pairs of dierent nature between the object and the contacting link, that restrict some or all the components of the relative velocities of the two bodies

5 System denition example (a) Four bar linkeage (b) Equivalent multiple cooperating limbs system system Figure: From closed chain mechanisms to multiple limbs cooperative robots : an illustrative example

6 equation system Recalling what stated in [1] : H( m x o x) = 0 o x = G T u o ẋ = G T u m x = J q mẋ = J q

7 Passive joints References Distinction had to be made between active and passive joint variables. A suitable permutation matrix P can be found that reorders joint angular/position variables q to have actuated joints on top, and unactuated joints at bottom: δ q T = (δq T A, δqt P )T = δq T P T where δq A (respectively, δq P ) is the q A (q P )-vector of actuated (unactuated) joint velocities. Correspondingly, the Jacobian matrix is partitioned as Jδq = JP 1 δ q = [ ] [ ] δq A J A J P δq P In order to avoid more complicate notation in the following analysis q will be indicated as q.

8 formulation Dening G = G H T J = H J t = H T t The mobility of the system is then studied by analyzing the constraint equation system [J A J P G T ] q A q Ṗ u = 0

9 All possible motions of the system belong to the nullspace (or kernel) of the constraint matrix [ J A J P G T ] and, hence, can be rewritten as linear combinations of vectors forming a basis of the nullspace. Mobility matrix parametrization By suitable linear algebra operations, such a basis can always be written in a block-partitioned form q A [ ] x 1 q Ṗ C11 C = Cx = 12 0 x 2 0 C 22 C 23 u x 3

10 q A q Ṗ u [ C11 C = Cx = C 22 C 23 ] x 1 x 2 x 3 redundancy indeterminancy coordinated motions C 23 ulterior partition [ C T 12, C T 22] T ulterior partition

11 A simple, simple example Structure Let 's examine the example reported below C1 (0 1 0) C2 (1 1 0) We have to nd: C 11 [ C T, ] 12 C T T 22 O1 (0 1 0) O2 (1 0 0) C 23

12 A simple, simple example Structure Let 's examine the example reported below C1 (0 1 0) C2 (1 1 0) We have to nd: C 11 [ C T, ] 12 C T T 22 O1 (0 1 0) O2 (1 0 0) C 23

13 A simple, simple example Structure Let 's examine the example reported below C1 (0 1 0) C2 (1 1 0) We have to nd: C 11 [ C T, ] 12 C T T 22 O1 (0 1 0) O2 (1 0 0) C 23

14 A simple, simple example Structure Let 's examine the example reported below C1 (0 1 0) C2 (1 1 0) We have to nd: C 11 [ C T, ] 12 C T T 22 O1 (0 1 0) O2 (1 0 0) C 23

15 According with the type of contact (hf w/h friction or soft nger) [ 0.3 ] C 1,1 = / C 1,2 = C 22 = C 2,3 = / or C 2,3 =

16 Parallel structure equivalence The analyzed example can be considered a 4 bar linkage "translation" (a) The four-bar linkage (b) (c) Figure: Parallel structure analysis

17 An interesting example References Santello's zip postures

18 An interesting example References Santello's zip postures C 11 = /

19 An interesting example References Santello's zip postures C 22purged =

20 An interesting example References rank(c 22 ) = 4. No permitted motions: