1 Introduction. Control 2:

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1 Introduction Kinematics of Wheeled Mobile Robots (No legs here. They ork like arms. We e seen arms already) For control purposes, the kinematics of heeled mobile robots (WMRs) that e care about are the rate kinematics. Of basic interest are to questions: a) Ho do desired motions of the robot translate into desired motions of the heels. b) Ho do measured motions of the heels translate into equialent motions of the robot.

2 Conentions Wheels normally hae up to to degrees of freedom (steer and drie) ith respect to the ehicle to hich they are attached. The relationship beteen heel angular elocity and the linear elocity of the contact point is: Conentions : ehicle, fixed to some point on the ehicle hose motion is of interest. s: steer, positioned at the hip/steer joint. Moes ith the boom. c: contact point, moes ith the contact point. Has orientation of heel in the plane. s c y x r Figure 4 Frames for WMR Kinematics. Figure 3 Wheel Angular and Contact Point elocity. To accomodate modelling of passie castors, e allo the steer axis to be potentially different from the the contact point. Define the folloing frames of reference: : orld, fixed to the enironment

3 3 Forard Rate Kinematics (Sensed Forard Solution) Gien the linear and angular elocity of the ehicle frame ith respect to the orld frame, the linear elocity of the contact point can be computed. Clearly: r c s r + r s +r () c Lets differentiate this in the orld frame. c c c d rs d rc s + ( ) + ( ) dt dt 0 s + s + r s + s r c s + r s + ( + s ) r c So, e also need to kno rate of rotation of the steer frame ith respect to the ehicle s. This is measureable or otherise knon. The () 3 Forard Rate Kinematics (Sensed Forard Solution) 3 other ectors and are knon ehicle dimensions. Express all ectors in ehicle coordinates: c r s and its no clear that e also need the steer/ castor angle to get R c. When the steer mechanism is oer the contact s point, align frames c and s and e hae r c 0 and the solution reduces to: Either case is of the form: cb. The notation means the elocity of frame a ith respect to b a frame b (i.e a expressed in the coordinates frame c.why such a screy notion? Contact point elocities are of the heel (a) ith respect to the orld (b) expressed in ehicle coordinates (c).. It is not unusual to find that expressing a ector equation in some coordinate system leads to a need to kno more things. That s hat the gyros are for in inertial guidance. r c s (3) c s + kˆ r s + c kˆ R crc + kˆ r (4) c (5) c h(, )

4 But, this relationship is actually linear in these ariables, so it is of the form: x y x (6) J y Where J is the forard elocity Jacobian. We can rite such an equation for each heel and easily compute the elocity of each heel contact point. If you ant to compute the angular elocity of the heel about its axle, you must first project the contact point elocity onto the forard heel direction: θ c c ŷ c (7) If the castor angle is passie and has stabilized, the heel ill be lined up ith its contact point elocity. If you ant to compute the required steer angle, just use the direction gien by the contact point elocity ector. 3 Forard Rate Kinematics (Sensed Forard Solution) 4 3.Example: Differential Steer This is the solution for the general (planar) case. 3. Example: Differential Steer The case of differential steer is pretty easy: Figure 5 Differential Steer. l y W x Let the to heel frames be called l and r for left and right. Let the forard elocity be and the angular elocity is. The dimensions are: r r W r W l r (8)

5 Equation (4) for each heel reduces to to scalar equations: r l + W ---- W ---- W W Here, the equation for the elocity in the sideays direction says it anishes (hen expressed in body coordinates), so e don t need to rite it. This eliminates one degree of freedom because there are only dof in body coordinates. e are therefore able to sole for the (normally 3 dof) motion using to measurements. (9) 4 Inerse Rate Kinematics (Actuated Inerse Solution) 5 3.Example: Differential Steer 4 Inerse Rate Kinematics (Actuated Inerse Solution) No for the opposite problem. Suppose e hae measurements of heel angular elocities θ c and steer angles. We ant to find the associated ehicle linear and angular elocity. Once again, the forard solution is: x y x J (0) y Which is in the form of an obserer. The unknons are x y T in this case There are three of them and only to constraints are proided per heel. Clearly, the linear elocity of one point on a rigid body does not constraint all 3 dof of planar motion. Therefore, e need at least to sets of heel (steer, speed) measurements in general. Then, the problem is oerdetemined and a best fit

6 solution is called for hich tolerates the ineitable inconsistency in the measurements. 4. Example: Differential Steer In body coordinates, the constraints for to heels generate to scalar equations. Equation (9) is easy to inert: r l W W W W r l () 5 Summary 6 4.Example: Differential Steer 5 Summary Rate knematics for heeled mobile robots are pretty straightforard in the general case in D. The inerse problem is often oerdetermined because the motions of n heels must ultimately be consistent ith 3 dof motion. This is soled like any oerdetermined system. 6 Notes Add something about skid steering 7 References

7 7 References 7 4.Example: Differential Steer

8 7 References 8 4.Example: Differential Steer