<p> Kinematics of Mobile Robot Platforms:</p><p>1) Differential Steer Robot: Assumptions: IC c A1. Planar surface A2. No slip: v vl u A3. Pure Roll v z r P y 2b</p><p> x 2) Tricycle Steer Robot: Assumptions A1. Planar Surface A2. No Slip A3. Pure Roll u</p><p>ICc  v z P</p><p> y 2b</p><p> x 3. Skid-steer Robot: Assumptions:</p><p>ICc A1. Planar Surface</p><p>IC ˙ l  l rll u</p><p> v V c ˙  r rr r z P</p><p>ICr </p><p> y 2b</p><p> x</p><p>Consider a model as shown in the figure below, consisting of 3 bodies, the chassis as the center body and the contact patch of each track as separate bodies connected to the chassis with prismatic joints. CP l x y RC b m CP C r</p><p>Y 2b b m l</p><p>X The prismatic joint motion is related to the wheel rotation through the following kinematics:</p><p>˙ ˙ (1) dl   rl r and ˙ ˙ (2) dr   l r r Now consider the model with instant centers identified for each of the three bodies: IC</p><p>IC l x y r Icl-A r A ICl r r C-A ICr C r C-B IC r r B Icl-B Y</p><p>X</p><p>The velocity of an arbitrary point on CPl, A is given as, ˙ (3) v A  vC  z  rCA  dl xˆ where rC-A is a vector from point C to A as shown in Fig. 3. Similarly, the velocity of an arbitrary point on CPr , B is given as, ˙ (4) v B  vC  z  rCB  d r xˆ where rC-B is a vector from point C to B as shown in Fig. 3. The velocity of points A and B can be described in terms of rotation about the instant centers associated with CPl, CPr respectively as, v    r (5) A z ICl A v    r (6) B z ICr B where rICl-A is a vector from the instant center of CPl to A and rICl-B, is a vector from the instant center of CPr to B as shown on the figure above. Combining equations 1, 3 and 5 for the left track, 2, 4, and 6 for the right track, and collecting terms through the distributive nature of addition in the cross product yields</p><p>0  v    r  r  d˙ xˆ C z CA ICl A l (7) 0  v    r  r  d˙ xˆ C z CB ICl B r (8) or  ˙ r xˆ  v    r  l l l C z ICl (9)  ˙ r xˆ  v    r  r r r C z ICr (10) T T Where rICl = (xic, ylic) , rICr = (xic, yric) , are vectors from the robot chassis centroid (C) to the instant T centers for CPl and CPr respectively and vC = (vx, vy) is the velocity of the robot at C. These equations are expanded into scalar components,</p><p>˙ vx  z ylic   r      l l l  (11) v y  z xic   0  ˙ vx  z yric   r       r r r  (12) vy  z xic   0  and then combined into the matrix expression, </p><p>vx    ˙  J v  J l 1  y  2  ˙  (13)    r  z  where </p><p>1 0  ylic  J  1 0  y  1  ric  (14) 0 1 xic </p><p>l rl 0  J   0  r  2  r r  (15) .  0 0  The direct kinematics of the tracked robot system is written in its usual form as,</p><p>vx    ˙  v  K l  y  eq  ˙  (16)   r  z  where</p><p>yricl rl  ylic r rr  1 K  J 1J   x  r  x  r  eq 1 2  ic l l ic r r  . (17) yric  ylic   l rl  r rr  4: Vehicle with Trailer castors:</p><p>ICc u  a 2   2 2 - 1 </p><p> v 90- 2</p><p> co2</p><p> c a1 </p><p> a  + 90 3 1 </p><p> y 1</p><p> co  1 x 1</p><p>Encoders report the orientation (1, 2) and rotation (1, 2) of each wheel. Based on these measurements, estimates (termed actual for the purposes of this paper) of the robot motion are given as:</p><p>T * * * * ˙ (1) Vu ,Vv , z   f  1 , 2 , 1 </p><p>With</p><p>* ˙ f1  1 , 2 , 1   Vu  ˙ r sin   a cos   cos  1 1 2 1  3 2 1   co1 sin 1   a3 cos 2     sin 2  1   (2) * ˙ f 2  1 , 2 , 1   Vv  ˙ r sin   a cos   sin  1 1 2 1  3 2 1  c1  co1 cos1   a3 cos 2     sin 2  1   (3) ˙ * ˙ 1r1 sin 2   1  f3  1 , 2 , 1     (4) a3 cos 2   </p><p>Where</p><p>2 2 a3  c1u  c2u   c1v  c2v </p><p>2 2   c1  co1 cos 1  c2  co2 cos 2    co1 sin 1  co2 sin 2  (5) and</p><p>  atan 2c2v  c1v ,c2u  c1u </p><p> atan 2 co2 sin 2  co1 sin 1 ,c2  co2 cos 2  c1  co1 cos 1  (6)</p><p>The geometric parameters, c1, c2, co1, co2 are shown in Fig. 4 and c1u, c1v, c2u, c2v are the (u,v) coordinates along c1, c2. </p>