ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

Transcription

1 ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 9 MODELING AND ANALYSIS OF WHEELED MOBILE ROBOTS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian Institute of Science Bangalore 56 12, India asitava@mecheng.iisc.ernet.in NPTEL, 21 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 1 / 88

2 ...1 CONTENTS...2 LECTURE 1 Wheeled Mobile Robots (WMR) on Flat Terrain...3 LECTURE 2 Wheeled Mobile Robots (WMR) on Uneven Terrain...4 LECTURE 3 Kinematics and Dynamics of WMR on Uneven Terrain...5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 2 / 88

3 OUTLINE...1 CONTENTS...2 LECTURE 1 Wheeled Mobile Robots (WMR) on Flat Terrain...3 LECTURE 2 Wheeled Mobile Robots (WMR) on Uneven Terrain...4 LECTURE 3 Kinematics and Dynamics of WMR on Uneven Terrain...5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 3 / 88

4 CONTENTS OF LECTURE Introduction to wheeled mobile robots. Different kinds of wheels and their modeling. Kinematics of wheeled mobile robots on flat surfaces. Modeling of wheel slip. Modeling of a WMR with omni-directional wheels. Simulation of a WMR with omni-wheels. Summary ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 4 / 88

5 INTRODUCTION Previous 8 modules, all robots considered were stationary with a fixed link or a base. Several robots are now designed to have mobility They can move on a surface, in water or in air. Only robots with capability of mobility on a surface considered. Earliest examples are automated guided vehicles (AGVs) with wheels used on flat factory floors. More recently autonomous robots with legs and/or a combination or wheels and legs (hybrid) have been built. Vast majority are wheeled mobile robots or WMRs as they are more efficient and faster than legged or tracked vehicles. Legged and tracked vehicles can navigate rough terrain more easily. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 5 / 88

6 INTRODUCTION Autonomous vehicles are used in Industrial environments to move material (parts and finished goods) from one place to another. Military and security applications. Hazardous environments such as inside nuclear reactors or in deep sea bed. Providing mobility to handicapped persons. Planetary exploration. Analysis of WMRs involve kinematics, dynamics, control, sensing, motion planning, obstacle avoidance... This lecture deals with kinematics and dynamics of WMRs moving on a flat surface or a plane. Next lecture deals with WMRs moving on uneven or rough terrains. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 6 / 88

7 EXAMPLES OF WMRS MARS Rover from NASA A WMR for moving on uneven terrain AGV s used on factory floors A WMR with omni-directional wheels Figure 1: Some Wheeled Mobile Robots ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 7 / 88

8 INTRODUCTION Muir and Newman (1987) A WMR is a robot capable of locomotion on a surface solely through the wheel assemblies mounted on it and in contact with a surface. A wheel assembly is a device which provides or allow relative motion between its mount and the surface on which it is intended to have single-point of rolling contact. In practical wheels, due to deformation, there is area contact. Different types of wheels used in WMRs 1) Conventional, 2) Omni-directional and 3) Ball wheels ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 8 / 88

9 INTRODUCTION Connection to robot body Actuating motor Conventional wheel Hub motor or actuator + transmission to wheelaxis Two rows of barrels in the wheel Encoder For connecting to robot body Spherical ball shaped wheel Drive mechanism complicated & not shown Figure 2: Three main types of wheels in WMR s Conventional wheel most commonly used Rotation about axis of wheel & steering. Omni-directional wheels also called Swedish wheels: Barrels on the periphery. Barrel can rotate at an angle to the wheel rotation axis 9 in figure. Barrel rotation is not actuated & barrel rotation leads to sliding of wheel. Two DOF in each wheel & steering. Ball or spherical wheels Essentially a sphere which can rotate about two axis. Complicated drive and rotation measuring arrangement. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 9 / 88

10 SINGLE WHEEL KINEMATICS Disk on a plane Configuration space (x,y,ϕ), ϕ is the steering angle 1, r is radius. {} wheel (disk) Axis of rotation rotation angle -θ x -θ y Normal to plane at point of contact r φ Path of contact point Tangent at (x,y) Wheel rolling without slip Motion only along tangent to path ẋ sinϕ ẏ cosϕ =. Velocity perpendicular to the tangent vector is zero. Non-holonomic constraint Cannot be integrated to obtain f (x,y,ϕ) = Constrains q = (ẋ,ẏ, ϕ) T, and not q = (x,y,ϕ) T. Figure 3: Wheel rolling on a plane 1 Tilting of wheel from the normal not considered and rotation of wheel θ not required for analysis. If thin disk, tilt must be considered. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 1 / 88

11 SINGLE WHEEL KINEMATICS Justification for non-holonomic nature Assume ẋ sinϕ ẏ cosϕ = can be integrated. This implies existence of f (x,y,ϕ) = Two DOF Choosing any two out of x,y,ϕ automatically determines the third! For example, at x = y = ϕ = ϕ is fixed 2 No other value(s) of ϕ is possible! This is clearly false Any ϕ is possible at (,) by simply rotating disk about the normal. Hence f (x,y,ϕ) = is not integrable. The kinematics of a single wheel is given by ẋ cosϕ ẏ = sinϕ v + ω ϕ 1 where v and ω are the forward speed and steering rate. 2 At most finitely many ϕ if f (x,y,ϕ) = is nonlinear. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

12 TWO WHEEL ROBOT BICYCLE Four generalised coordinates Y (x,y) Point of contact of rear φ WMR wheel. C ψ Orientation of body & rear wheel with X axis. R l ϕ Steering angle of front ψ front wheel wheel with respect to body Two rolling constraints in each wheel: x y rear wheel ẋ sinψ ẏ cosψ = {} C Instantaneous X ẋ sin(ψ + ϕ) ẏ cos(ψ + ϕ) l ψ cosϕ = Centre Kinematics equations Figure 4: (bicycle) Two wheel robot ẋ ẏ ψ ϕ = cosθ cosϕ sinθ cosϕ sinϕ/l v f + Lines perpendicular to front and rear wheel velocity vector meet at C. Motion of the bicycle is instantaneously rotation about C. 1 ω ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

13 THREE WHEEL ROBOT TRI-CYCLE Y φ Front wheel is steered. WMR Speeds of rear wheels are C different when making a turn. Wheel speed v i = θ i r i, i = 1,2,3, R l r ψ i is radius of wheel i. front wheelone rear wheel is driven and the speed of the second rear wheel rear wheels must adjust so that there is a single instantaneous centre C x y (the second rear wheel can be {} X free or a differential is used). C Instantaneous Centre Figure 5: Three wheeled mobile robot or tri-cycle Kinematics of tri-cycle similar to bicycle. C determine relation between R, θ i, r i, l, ψ and ϕ. If no single C Wheel(s) slip will occur. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

14 C CAR LIKE MOBILE ROBOTS & OTHERS d l φi φ o C Instantaneous Centre Figure 6: robot Four-wheeled mobile Four-wheeled (car like) mobile robot Steering angle of front wheels is different in a turn. Ackerman steering to avoid slippage The steering angle of inside wheel ϕ i and outside wheel ϕ o are related as cot(ϕ i ) cot(ϕ o ) = d/l Multi-axle vehicles: truck + trailer Steering front wheel to the left results in right motion of wheels on third axle Non-minimum phase system. More difficult to analyse and model. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

15 OMNI-DIRECTIONAL WHEELS Two rows of barrels Six equally spaced free barrels on the periphery. Each barrel can rotate about its axis as shown. Two rows of barrel One barrel always in contact with ground. Distance of point of contact from the vehicle centre changes. Rotation speed is θ & Sliding speed is σ. Barrel rotation axis Barrel shaped rollers Figure 7: Omni-directional Wheel (Balakrishna & Ghosal, 1995) Two components of velocity of wheel centre for a general case of barrel axis at an angle α to wheel axis (r θ + σ cosα,σ sinα) T. α = 9 Components are (r θ,σ) T, r is radius of wheel. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

16 Y σ 2 v 2 WMR WITH THREE OMNI-WHEELS v1 WMR with omni-wheels of radius r. x 12 o 12 v {} σ 3 3 X y o ψ Figure 8: WMR with omni-directional wheels (Balakrishna & Ghosal, 1995) L σ 1 No steering wheel, wheel rotation speed θ i, i = 1,2,3. Kinematics No slip condition at wheels θ 1 θ 2 θ 3 = 1 r 1 L 1 3/2 1/2 L 2 3/2 1/2 L3 = [ R ] ẋ ẏ ψ L i, i = 1,2,3 are the contact points. 3 3 matrix [R] (analogous to manipulator inverse Jacobian) can be inverted WMR controllable with θ i. Can obtain sliding speeds (σ 1,σ 2,σ 3 ) T in terms of (ẋ,ẏ, ψ) T Not invertible WMR cannot be controlled by σ i, i = 1,2,3 alone. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88 ẋ ẏ ψ

17 MODELING OF SLIP For rolling without slip in a conventional wheel of radius r, wheel centre velocity v and wheel angular velocity ω are related by v = rω. Either wheel and/or ground must deform for generating tractive force to drive a wheel. Deformable wheel and/or ground combination Wheel slip will occur (Shekhar, 1997). Wheel slip is defined as λ = ( θ ω )/y where ω = v/r and θ is the wheel angular velocity. y = ω when ω > θ and y = θ when ω < θ Above implies 1 λ 1 Pure rolling λ = Rolling in place λ = 1 Skidding λ = 1 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

18 MODELING OF SLIP (CONTD.) Deformable Wheel Rigid ground Figure 9: dynamics Single wheel N θ τ F t. x Translational and rotation dynamics F t = M w ẍ, τ = J w θ + F t r F t : tractive force at wheel-ground interface. τ: torque applied at wheel axle. M w, J w : mass and inertia of wheel. ẍ, θ: linear and angular acceleration. In state-space form ẋ = f(x) + gτ Lie algebra approach Not locally controllable if tractive force F t is constant. F t must be a function of ẋ and θ. F t is a function of normal reaction N and adhesion coefficient µ a. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

19 MODELING OF SLIP (CONTD.) µ a peak 1. µ a λ, wheel slip 1. Figure 1: µ apeak Typical adhesion coefficient Vs wheel slip Typically µ a is a function of wheel slip λ and has a maximum µ a peak. Typical plot shown in figure (Dugoff et al. 197) Tractive force developed F t = µ a (λ)n Actually < F t µ a N: Proper sign of F t for acceleration or braking. Stable region Increasing µ a with wheel slip. After µ a peak, tractive force falls with increasing wheel slip Unstable! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

20 EQUATIONS OF MOTION Equations of motion using Lagrangian formulation (see Balakrishna & Ghosal, 1995) Kinetic energy of platform of mass M p and inertia I p. Kinetic energy of wheels of inertia I i, i = 1,2,3. No potential energy as motion on flat plane. Including tractive forces at wheels F ti, torque at each wheel τ i and an approximate µ a Vs λ curve. Equations of motion 6 ODE s to take into account slip M p M p I p ẍ ÿ ψ + Ψ I 1 I 2 I 3 M p M p θ 1 θ 2 θ 3 + r F t1 F t2 F t3 ẋ ẏ ψ = r [ R ] T = τ 1 τ 2 τ 3 F t1 F t2 F t3 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 2 / 88

21 CONTROL Desired Cartesian path X d = (x d (t),y d (t),ψ d (t)) T prescribed. PID control F = [K v ] (Ẋd Ẋ) + [K p] (X d X) + [K i ] (X d X) dτ X = (x(t), y(t), ψ(t)) T. Cartesian forces F is related to (wheel) actuator torque by τ = [ R ] T 1 F Model based control scheme (see Module 7, Lecture 3) using ideal rolling τ = [α] τ + β where [α] = [ R ] T [M ], β = [ R ] T { Ψ [ Q ] Ẋ} τ = Ẍd + K v (Ẋd Ẋ) + K p(x d X) where [M ] = ([M] + [ R ] T [I ] [ R ]) is the mass matrix corresponding to ideal rolling. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

22 NUMERICAL SIMULATION RESULTS Geometry of WMR: L 1 = L 2 = L 3 =.5 m, r =.1 m Inertia parameters: M p = 5. Kg, I p = 5. kg m 2, I i = 2. kg m 2. Controller gains equal: K pi = 15., K vi = 2 K pi, and K ii =.1. Figure 11: Straight line trajectory control using PID controller (Balakrishna & Ghosal, 1995) Figure 12: Circular trajectory control using model-based controller (Balakrishna & Ghosal, 1995) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

23 NUMERICAL SIMULATION RESULTS (CONTD.) Performance of PID controller is quite poor. Maximum x error is approximately.5 m and maximum y error is approx.2 m. Model based controller using ideal rolling as model performs slightly better. As the µ a peak decreases from.8 to.8, performance of model based controller becomes poorer. Maximum radial error from desired circular trajectory of radius 2. m is approximately 1.7 m. It is important to model or take into account slip in WMR models. This is also borne out by experiments! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

24 SUMMARY OF MODELING OF WMR WMR modeling and analysis is very different from serial and parallel manipulators. Base is not fixed. Wheel-ground contact results in non-holonomic constraints Compare with holonomic in joints. Non-holonomic constraints does not restrict the configuration space but restrict space of velocities! Various kinds of wheels in use Conventional, omni-directional and ball wheels. Conventional wheels rotate about the wheel axis and can be steered about the normal. Omni-directional wheel can rotate about its axis, slide along another direction and also steered. Ball wheels can rotate about two different axis. WMR with three wheels simplest Four wheeled and multi-axle WMR s more difficult to model and analyse. Slip, due to deformation, always present in wheels Need to be taken into account in WMR dynamics and control. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

25 OUTLINE...1 CONTENTS...2 LECTURE 1 Wheeled Mobile Robots (WMR) on Flat Terrain...3 LECTURE 2 Wheeled Mobile Robots (WMR) on Uneven Terrain...4 LECTURE 3 Kinematics and Dynamics of WMR on Uneven Terrain...5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

26 CONTENTS OF LECTURE Introduction Modeling of torus-shaped wheel and uneven terrain Single wheel on uneven terrain Kinematic and dynamic modeling and simulation. A WMR for traversing uneven terrain without slip. Summary ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

27 INTRODUCTION Most WMR are used in industrial environments Flat & structured surfaces. Recent interest in uneven and rough terrains & off-road environments. Planetary exploration. DARPA Grand Challenge to develope a fully autonomous ground vehicle capable of completing a off-road course in limited time (see link 3 for more details). Luxury cars. Flat terrain Vehicle platform has 3 DOF consisting of position (x,y) and orientation ψ Uneven terrain Vehicle platform can possibly have all three components of translation and three components of orientation Up to 6 DOF. 3 Copy and paste link in a New Window/Tab by right click. Close New Window/Tab after viewing. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

28 INTRODUCTION A P Figure 13: terrain Wheel slip on uneven B Q Two wheels connected by a fixed length axle AB. Points of contact with uneven ground P, Q can change Length PQ is variable and not equal to AB. Variation of length PQ require a velocity component along axle AB and along the normal (Waldron, 1995). No instantaneous centre compatible with both wheels Wheel slip will occur. Wheel slip leads to a) localization errors, and b) wastage of fuel. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

29 INTRODUCTION To overcome wheel slip Variable length axle (Choi & Sreenivasan, 1999). Add passive prismatic joint in axle Prismatic joint changes axle length by required amount to ensure compatible instantaneous centre for both wheels. At large inclination, gravity causes prismatic joint to change length in undesired way! Actuated (and controlled) prismatic joint Accurate sensing of slip is required. New concept of WMR capable of traversing uneven terrain without slip. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

30 INTRODUCTION Main concepts (see Chakraborty & Ghosal (24, 25)). Use of torus shaped wheel Wheel has single point contact with uneven terrain. Torus shaped wheels connected to WMR body with passive rotary joints. Passive rotary joint allow lateral tilting and wheel-ground contact distance (PQ) to change. Three actuated joints (for a 3DOF model) in WMR. Rear wheels are driven and can tilt laterally. Front wheel steered and can roll freely. Use of contact equations (Montana, 1988) to model wheel-ground contact. WMR modeled as a parallel robot at each instant. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 3 / 88

31 ˆX MODELING OF SURFACE Surface {} Ẑ p fv n Ŷ Figure 14: A surface in R 3 fu Tangent Plane at p v (u, v) u A surface in R 3 can be represented in parametric form (x,y,z) T = f(u,v), (u,v) U R 2 maps to p = (x,y,z) T on the surface. At a point, tangent plane defined by vectors f u = f u and f v = f v. The normal to the surface is n = f u f v f u f v The vectors f u f u, f v f v, and n form a right-handed basis a at p. a If f u and f v are not orthogonal, then the orthogonal set is {f u / f u,n f u / f u,n}. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

32 {} MODELING OF A TORUS-SHAPED WHEEL Uneven Terrain vw = constant p Torus uw = constant r 2 v g = constant u g = constant Figure 15: A torus-shaped wheel on uneven terrain r 1 Equation of a torus in parametric form x = r 1 cosu w y = cosv w (r 2 + r 1 sinu w ) z = sinv w (r 2 + r 1 sinu w ) r 1 and r 2 are the two radii associated with a torus. Subscript w on the parameters u and v denote a wheel. Uneven terrain/ground (x,y,z) T = f(u g,v g ) ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

33 METRIC, CURVATURE AND TORSION OF A SURFACE From the parametric equation, define Second partial of f, f ( )( ) = 2 f, with respect to u and ( ) ( ) v, Partial of n, n ( ), with respect [ to u and ] v. fu Metric on a surface [M] = f v Curvature form [K] = f u n u f u 2 f u n v f u f v f v n u f u f v f v n v f v 2 [ ] fv f uu f v f uv Torsion form [T] = f u 2 f v f v 2 f u Metric defines distance, Curvature determines in-plane bending and Torsion determines out-of-plane bending on a surface. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

34 MODELING OF A TORUS-SHAPED WHEEL {} Uneven Terrain Torus uw = constant vw = constant r 1 r 2 p v g = constant u g = constant For the torus-shaped wheel [ ] r1 [M w ] = r 2 + r 1 sinu w [ ] 1 r [K w ] = 1 [T w ] = [ sinu w r 2 +r 1 sinu w cosu r 2 + r 1 sinu w ] Figure 16: A torus-shaped wheel on uneven terrain ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

35 MODELING OF UNEVEN TERRAIN Uneven terrain Smooth and hard, not terrains with sand, dirt or any discontinuities! Explicit or parametric equation not available. Local elevation of a point is known from measurement (laser scanner). Ill-posed problem to obtain f(u, v) from measurements Non-uniqueness. Use bi-cubic and B-spline surfaces (see Mortenson, 1985) Bi-cubic surface patch: f(u,v) = 3 i= 3 j= a iju i v j, (u,v) [,1] Can be determined from 4 corner points of the patch. Patches can be smoothly connected to make up the whole surface. Higher-order continuity can be obtained by using Non-uniform Rational B-Spline (NURBS) MATLAB R Spline Toolbox Partial derivatives of surface available to compute metric, curvature and torsion form. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

36 EXAMPLES OF UNEVEN TERRAINS Figure 17: A bi-cubic uneven surface Figure 18: A B-spline uneven surface ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

37 KINEMATICS OF CONTACT Surface 1 {} p {C r1 } {C l1 } Surface 2 {C l2 } {C r2 } Figure 19: Two arbitrary surfaces in single-point contact Two surfaces 1 and 2 described with respect to {C r1 } and {C r2 }. Parametric equations are f(u 1,v 1 ) and f(u 2,v 2 ). At point of contact p, fix frames {C l1 } and {C l2 }. ψ Angle between X axis of {C l1 } and {C l2 }. (u 1,v 1 ), (u 2,v 2 ) and ψ define the 5 DOF for single-point contact. Define metric, [M], Curvature [K] and Torsion [T ] for the two surfaces at p. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

38 KINEMATICS OF CONTACT Contact equations: Relationship between ( u 1, v 1, u 2, v 2, ψ) and the linear and angular velocity components v x,v y,v z and ω x,ω y,ω z ( u 1, v 1 ) T = [M 1 ] 1 ([K 1 ] + [K ]) 1 [( ω y,ω x ) T [K ](v x,v y ) T ] ( u 2, v 2 ) T = [M 2 ] 1 [R ψ ]([K 1 ] + [K ]) 1 [( ω y,ω x ) T + [K 1 ](v x,v y ) T ] ψ = ω z + [T 1 ][M 1 ]( u 1, v 1 ) T + [T 2 ][M 2 ]( u 2, v 2 ) T = v z where the relative curvature of surface 2 with respect to 1 is [K ] = [R ψ ][K 2 ][R ψ ] T and the rotation matrix [R] is ( ) cosψ sinψ [R ψ ] = sinψ cosψ ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

39 KINEMATICS OF CONTACT Can also invert the equations (v x,v y ) T = [M 1 ]( u 1, v 1 ) T + [R ψ ][M 2 ]( u 2, v 2 ) T (ω y, ω x ) T = [K 1 ][M 1 ]( u 1, v 1 ) T [R ψ ][K 2 ][M 2 ]( u 2, v 2 ) T ω z = ψ [T 1 ][M 1 ]( u 1, v 1 ) T [T 2 ][M 2 ]( u 2, v 2 ) T v z = v z = holonomic constraint Ensures surfaces stay in contact! Five other equations need to be numerically integrated with initial conditions for solution. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

40 KINEMATICS OF CONTACT Contact equations similar to constraint equations for joints (see Module 2, Lecture 2) Main difference: equations contain derivatives with respect to time! Two main types of contacts : Pure rolling v x = v y = Important for WMRs. Pure sliding ω x = ω y =. Pure rolling v x = v y = v z = Three DOF in velocities. Very much unlike a 3 DOF spherical joint S joint x = y = z are also same for both links! Pure rolling Non-holonomic Only v x = v y = (also v z = ) and x, y, z coordinates of the contact point can change as the rolling proceeds! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 4 / 88

41 SINGLE WHEEL ON UNEVEN TERRAIN Torus shaped wheel {} Uneven Terrain Yw p w {3} {2} p {4} Z 4 C {w} r 2 {1} r 1 Zw X w {C r2 } is same as {}. {C r1 } is fixed at the wheel centre C, same as {w}. {C l1 } and {C l2 } at the contact point are denoted by {2} and {1}. {3} and {4} as shown in figure. Figure 2: Single wheel on uneven ground ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

42 KINEMATIC ANALYSIS 1 [T ] = 2 3 [T ] = 4 4 transformation matrices from assigned frames: l 1 m 1 n 1 u g l 2 m 2 n 2 v g l 3 m 3 n 3 f (u g,v g ), 1 2 [T ] = 1 l i, m i, n i, i = 1,2,3 are the components of {f u / f u,n f u / f u,n}. Transformation matrices from {2} to {4} are sinu w cosu w 1 cosu w sinu w r 1 1, 3 4 [T ] = cosψ sinψ sinψ cosψ 1 1 Transformation matrix from {4} to {w} is 1 4 w [T ] = sinv w cosv w cosv w sinv w r 2 1 ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

43 KINEMATIC ANALYSIS The transformation matrix from {w} to {} is w [T ] = 1 [T ] 1 2[T ] 2 3[T ] 3 4[T ] 4 w [T ] The contact equation for a single-wheel rolling without slip ( u w, v w ) T = [M w ] 1 ([K w ] + [K ]) 1 [( ω y,ω x ) T ] ( u g, v g ) T = [M g ] 1 [R ψ ]([K g ] + [K ]) 1 [( ω y,ω x ) T ] ψ = ω z + [T w ][M w ]( u w, v w ) T + [T g ][M g ]( u g, v g ) T = v z w denotes wheel and g denotes ground. Inputs: ω x, ω y, ω z. Integrate contact equations to obtain evolution of point of contact on wheel and ground in time. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

44 KINEMATIC ANALYSIS 1.6 Lateral shift of contact point on Uneven terrain 1.55 u(radian) time(sec) v/ψ v ψ(radian) Simulation for bi-cubic surface shown earlier. r 1 =.5 m, r 2 =.25 m. Wheel tilts as it rolls time Figure 21: Variation of u w,v w and ψ with t ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

45 KINEMATIC ANALYSIS.8.6 Contact Point Wheel Center yc/vg (m) Trace of wheel centre and ground contact point is different. Unlike a disk rolling on flat surface xc/ug (m) Figure 22: point Plot of wheel centre and ground contact ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

46 DYNAMIC ANALYSIS Equations of motion of a torus-shaped wheel on uneven terrain using Lagrangian formulation. Non-holonomic constraints (v x,v y ) T = [M w ]( u w, v w ) T +[R ψ ][M g ]( u g, v g ) T = (,) T after rearranging [Ψ(q)] q = Kinetic energy of wheel Angular velocity w [Ω] from rotation matrix as w [Ṙ] w [R] T. Linear velocity by differentiating position of wheel centre V w = ṗ w Kinetic energy: KE = 1 2 ΩT [I w ]Ω m w V w 2 Potential energy: PE = m w gz wc Equations of motion (see Module 6, Lecture 1) [M(q)] q + [C(q, q)] q + G(q) = τ + [Ψ(q)] T λ ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

47 DYNAMIC ANALYSIS Wheel dimensions: r 1 =.5 m, r 2 =.25 m. Mass of wheel m w = 1. kg. Inertia components of a torus-shaped wheel 1 4 m w (3r r 2 2 ) [I w ] = 1 8 m w (5r r 2 2 ) 1 8 m w (5r r 2 2 ) Initial conditions satisfy non-holonomic constrains. No external force τ = Torus-shaped wheels rolls down under gravity on surface shown next. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

48 DYNAMIC ANALYSIS 8 7 Contact Point Wheel Center yc/vg (m) 5 4 f(ug,vg) ug(m) vg(m) Figure 23: B-spline surface used for single wheel dynamics xc/ug (m) Figure 24: Trace of wheel centre and ground contact points ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

49 DYNAMIC ANALYSIS u(radian) psi(radian) time(sec) v(radian) time(sec) 3.5 vx/vy (m/s) 4 x X component of slip velocity Y component of slip velocity time(sec) Figure 25: with t Variation of u w,v w and ψ Slip components are very small (1 8 m/sec) time(sec) Figure 26: Components of slip velocity at wheel-ground contact point Simulation checked for conservation of energy. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

50 A WMR FOR TRAVERSING UNEVEN TERRAIN Three torus-shaped wheels connected to rigid platform with rotary (R) joints. Two possible configurations Platform with 3 DOF Each wheel attached to platform with two R joints. For rear wheels One R joint is actuated by a motor making the wheel roll. For front wheel One R joint represents steering. For rear wheels One R joint is passive allowing lateral tilting about axis perpendicular to wheel rotation axis. For front wheel One R joint represents free rolling of the wheel. WMR platform with 6 DOF Each wheel attached to platform by three R joints. 2 R joints in rear and front wheel function as above. Additional R joint in rear wheel allow for steering. Additional R joint in front wheel allow lateral tilt. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 5 / 88

51 3 DOF CONFIGURATION θ 2 δ 2 C 3 P C 2 C 1 θ 3 φ 3 θ 1 δ 1 Top Platform Center of Top Platform G 3 O {} G 2 G 3 DOF "Non holonomic" Joint 1 Figure 27: Schematic of a three-wheeled mobile robot with 3 DOF Wheel-ground contact has 3 DOF instantaneously Only velocities are restricted Non-holonomic joint. Grübler criterion DOF = 6(N J 1) + J i=1 F i N = 8, J = 9, and J i=1 F i = 15 DOF = 3. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

52 6 DOF CONFIGURATION δ 2 C 3 φ 2 P C 2 δ 3 φ 3 δ 1 C 1 φ1 Top Platform Center of Top Platform {} O θ 2 θ 3 θ 1 G 3 G G DOF "Non holonomic" Joint Figure 28: Schematic of three wheeled mobile robot with 6 DOF Wheel-ground contact 3 DOF non-holonomic joint. Grübler criterion DOF = 6(N J 1) + J i=1 F i, N = 11, J = 12 and J i=1 F i = 18 DOF = 6. Study kinematics and dynamics of 3 DOF configuration. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

53 SUMMARY WMR with fixed length axle, moving on uneven terrain, can slip. Variable length axle concepts and concept of passive tilting. Geometric modeling of torus-shaped wheel and uneven terrain. Contact equations representing 5 DOF between two surfaces in single point contact. Kinematic and dynamic analysis and simulation of single wheel on uneven terrain. Configuration of a three-wheeled mobile robot for traversing uneven terrain without slip. Kinematic, dynamic and stability analysis in next lecture. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

54 OUTLINE...1 CONTENTS...2 LECTURE 1 Wheeled Mobile Robots (WMR) on Flat Terrain...3 LECTURE 2 Wheeled Mobile Robots (WMR) on Uneven Terrain...4 LECTURE 3 Kinematics and Dynamics of WMR on Uneven Terrain...5 ADDITIONAL MATERIAL Problems, References and Suggested Reading ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

55 CONTENTS OF LECTURE Kinematic analysis a three-wheeled mobile robot Solution of the direct kinematics problem. Solution of the inverse kinematics problem. Formulation of Equations of Motion for Dynamic analysis. Simulation results. Stability of three wheeled mobile robot on uneven terrain Summary ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

56 KINEMATIC ANALYSIS θ 2 δ 2 C 3 P C 2 C 1 θ 3 φ 3 θ 1 δ 1 Top Platform Center of Top Platform G 3 O {} G 2 G 3 DOF "Non holonomic" Joint 1 Figure 29: Equivalent instantaneous parallel manipulator Instantaneous parallel manipulator with a platform connected to ground by three serial chains. Actuated: Rear wheel rotations, θ 1 and θ 2, front wheel steering ϕ 3. Passive: Rear wheel tilt δ 1, δ 2 and front wheel rotation θ 3. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

57 KINEMATIC ANALYSIS Analyse the WMR as a parallel manipulator at every instant. Instantaneously Wheels are not fixed as in a parallel manipulator! Non-holonomic (no slip) constraints and hence kinematics in terms of joint rates! Direct Kinematic: Given actuation rates θ 1, θ 2 and ϕ 3, the terrain and WMR geometry, find orientation of top platform p[r] and the position vector of the centre of the platform. Inverse kinematics: Given geometry of WMR and terrain and given any three of V px,v py,v pz, Ω px,ω py,ω pz, find rear wheel actuator inputs θ 1 and θ 2 and the steering input to the front wheel ϕ 3. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

58 KINEMATIC ANALYSIS: ALGORITHM Step 1: Generate the uneven terrain surface: Use bi-cubic patches or B-splines to reconstruct the surface from elevation data. Find [M], [K] and [T] for the ground and wheels at the three wheel-ground contact points. Step 2: Form contact equations: For each wheel, obtain 5 ODEs in u i, v i, u gi, v gi, and ψ i i = 1,2,3. For no-slip motion, set v x = v y = for each of the three wheels. ω x, ω y, and ω z are related to Ω px, Ω py, Ω pz, and the input and passive joint rates (ω x,ω y,ω z ) T = (Ω px,ω py,ω pz ) T + ω input Above equation couples all five sets of ODEs resulting in a set of 15 ODEs in 21 variables 15 contact variables, θ 1,θ 2,θ 3, δ 1,δ 2, and ϕ 3. Out of 21 variables, θ 1, θ 2, and ϕ 3 are actuated and known. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

59 KINEMATIC ANALYSIS: ALGORITHM Step 3: Obtain the angular and linear velocities of centre of the platform: If γ, β, α be a Z-Y-X Euler angle parametrization (see Module 2, Lecture 1, Slide # 12) representing the orientation of the platform, then Ω px = α cosβ cosγ β sinγ = f 1 (u i,v i,u gi,v gi,ψ i, u i, v i, u gi, v gi, ψ i ) Ω py = α cosβ sinγ + β cosγ = f 2 (u i,v i,u gi,v gi,ψ i, u i, v i, u gi, v gi, ψ i ) Ω pz = γ α sinβ = f 3 (u i,v i,u gi,v gi,ψ i, u i, v i, u gi, v gi, ψ i ), i = 1,2,3 If x c, y c, and z c denote the coordinates of the centre of the platform in {}, the linear velocity of the centre of the platform is (V px,v py,v pz ) T = ( xc, y c, z c ) T = V wi + (Ω px,ω py,ω pz ) T p ci i = 1,2,3 denote three wheels, p ci locates the point of attachment of the wheel to the platform from the centre of the platform, and V wi is the velocity of the centre of the wheel. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

60 KINEMATIC ANALYSIS: ALGORITHM Step 4: Form holonomic constraint equations: Distance between the three points C 1, C 2, and C 3 must remain constant for the moving platform to be rigid. Holonomic constraint are p C1 p C2 2 = l 2 12 p C2 p C3 2 = l 2 23 p C3 p C1 2 = l 2 31 p Ci, i = 1,2,3 locate C 1, C 2, C 3, from the origin of {} and l ij is the distance between centres of wheels i and j, respectively. Holonomic constraints are same as spherical-spherical pair (S-S) joint constraint of Module 2, Lecture 2, Slide # 34. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 6 / 88

61 KINEMATIC ANALYSIS: ALGORITHM Solution of Direct Kinematics Problem: Steps 1, 2 & 4: 15 ODEs, 3 algebraic equations in 21 variables. 18 Unknown variables θ 1, θ 2 and ϕ 3 are given! Differentiate holonomic constraints Convert to a system of 18 ODEs. Integrate using ODE solver with initial conditions. Obtain position vector of centre and orientation of platform from 21 variables at each t Solution of Inverse Kinematics Problem: Steps 1 to 4: 21 ODEs, 3 algebraic equations. Assuming linear velocity of the platform x c, y c, angular velocity about vertical, γ, are given 24 unknowns. Convert DAE s to ODEs. Integrate set of 24 ODEs and obtain required θ 1, θ 2 and ϕ 3 as function of t. Initial conditions for the direct and inverse kinematics problems must satisfy the holonomic and non-holonomic constraints. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

62 NUMERICAL SIMULATIONS RESULTS FOR DIRECT KINEMATICS Geometry of the WMR Length of the rear axle 1 m. Distance of the centre of front wheel from middle of the rear axle 1 m. Torus-shaped wheel: r 1 =.5 m, r 2 =.25 m. WMR centre is at the centroid of the triangular platform. Initial conditions: u 1 = , v 1 = 3π 2, u g 1 = 4.89 m, v g1 =.3917 m, ψ 1 = , u 2 = 1.556, v 2 = 3π 2, u g2 = 3.1 m, v g2 =.4 m, ψ 2 = , u 3 = , v 3 = 3π 2, u g 3 = m, v g3 = m, ψ 3 = 3.144, θ 3 =, δ 1 =, and δ 2 = All angles in radians. Initial conditions satisfy constraints. Actuator inputs: θ 1 = 1 rad/sec, θ 2 =.9 rad/sec, ϕ 3 =.5t rad/sec. Uneven surface same as used for single wheel dynamic simulation. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

63 4.5 4 NUMERICAL SIMULATIONS RESULTS FOR DIRECT KINEMATICS Wheel 3 Wheel Ground Contact Point Wheel Center Center of Platform.5.45 delta1 delta Wheel 2 Wheel δ 1,δ 2 (radians) Figure 3: Locus of wheel-ground contact point, wheel centre and platform centre time(sec) Figure 31: Variation of lateral tilt δ 1 and δ 2 The locus of wheel centres not same as wheel-ground contact point Uneven terrain and lateral tilt! Lateral tilt changes at different points of the uneven terrain. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

64 Constraint NUMERICAL SIMULATIONS RESULTS FOR DIRECT KINEMATICS 1.5 x Constraint 12 Constraint 13 Constraint time(sec) Figure 32: constraints Satisfaction of holonomic Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) 1 x x 1 16 time (sec) x 1 16 time (sec) time (sec) Figure 33: Slip velocities at wheel-ground contact points for three wheels The holonomic constraints are satisfied up to 1 7 m There is virtually no slip WMR traverses uneven terrain without slip!! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

65 NUMERICAL SIMULATIONS RESULTS FOR DIRECT KINEMATICS Figure 34: Video of direct kinematics of WMR See video 4 for a simulation of the three wheeled mobile robot on uneven terrain. 4 Copy and paste link in a New Window/Tab by right click. Close New Window/Tab after viewing. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

66 NUMERICAL SIMULATIONS RESULTS FOR INVERSE KINEMATICS Geometry of the WMR Same as used in direct kinematics Length of the rear axle 1 m. Distance of the centre of front wheel from middle of the rear axle 1 m. Torus-shaped wheel: r 1 =.5 m, r 2 =.25 m. WMR centre is at the centroid of the triangular platform. Initial conditions: u 1 = 1.581, v 1 = 3π 2, u g 1 = m, v g1 =.4895 m, ψ 1 = , u 2 = , v 2 = 3π 2, u g2 = 2 m, v g2 =.5 m, ψ 2 = , u 3 = 1.588, v 3 = 3π 2, u g 3 = m, v g3 = m, ψ 3 = , z c = 2.22 m, α =.448, and β =.498 All angles in radians. Initial conditions satisfy constraints. Given inputs: ẋ c =.3 m/sec, ẏ c =.15 m/sec, and γ =.5t rad/sec. Uneven surface same as used for direct kinematics. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

67 3.5 3 NUMERICAL SIMULATIONS RESULTS FOR INVERSE KINEMATICS Wheel ground Contact point Center of Platform Wheel Center.4.35 Variation of delta1 and delta2 delta1 delta δ 1,δ 2 (radians) Figure 35: Locus of wheel-ground contact point, wheel centre and platform centre time(sec) Figure 36: Variation of lateral tilt δ 1 and δ 2 Due to uneven terrain, locus of wheel centres are not same as wheel-ground contact point. Lateral tilt changes at different points of uneven terrain. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

68 Constraint NUMERICAL SIMULATIONS RESULTS FOR DIRECT KINEMATICS 3 x 1 7 Constraint 12 Constraint 13 Constraint time(sec) Figure 37: constraints Satisfaction of holonomic Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) 1 x x 1 16 time (sec) x 1 16 time (sec) time (sec) Figure 38: Slip velocities at wheel-ground contact points for three wheels The holonomic constraints are satisfied up to 1 7 m There is virtually no slip WMR traverses uneven terrain without slip!! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

69 DYNAMIC ANALYSIS Formulation of equation of motion for WMR using Lagrangian formulation. Kinetic energy of wheels, platform and links connecting actuated and passive joints to platform. Potential energy due to gravity. 15 contact variables at three wheel-ground contact points, 3 passive, 3 actuated and 6 variables for position and orientation of WMR platform Total 27 generalised coordinates. Three actuating torques, two in rear wheel and for front wheel steering. Need 24 independent constraint equations for the system to be well-posed Inverse kinematics equations!! Derive equations of motion subjected to holonomic and non-holonomic constraints (see Module 6, Lecture 1). ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

70 LAGRANGIAN FORMULATION Total kinetic energy KE = (KE) w1 +(KE) w2 +(KE) w3 +(KE) platform +(KE) actuators +(KE) links Total potential energy PE = (PE) w1 +(PE) w2 +(PE) w3 +(PE) platform +(PE) actuators +(PE) links All kinetic energy and potential energy components can be found (see Chakraborty & Ghosal, 25). Constraints equation from inverse kinematics: [Ψ] q = [Ψ] is a matrix. Equations of motion from Lagrangian formulation [M(q)] q + [C(q, q)] q + G(q) = τ + [Ψ(q)] T λ [M(q)] is a mass matrix, [C(q, q)] q, G(q) and τ are 27 1 vectors. Only three elements of τ, corresponding to θ 1, θ 2, ϕ 3, are non-zero. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 7 / 88

71 LAGRANGIAN FORMULATION λ is 24 1 vector of the Lagrange multipliers. Solve for λ as shown in (see Module 6, Lecture 1, Slide # 17). Finally obtain a set of 27 second-order ODEs Obtained in symbolic form by using Mathematica R extensively! With actuators locked and wheels tilted, WMR falls under own weight! Contrary to a normal parallel manipulator!! Wheel-ground contact modeled as instantaneous 3 DOF joint Not like a spherical (S) joint. Form closure not present Additional terms modeling a torsion springs and a damper (preventing falling under own weight with actuators locked) is added in τ corresponding to lateral tilts k si δ i + k di δi, (i = 1,2) k si and k di are spring constant and damping. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

72 DYNAMIC ANALYSIS: ALGORITHM Step 1: Generate the uneven terrain surface Reconstruct the surface from elevation data. Derivative of constraints are required C 3 continuity required. Fourth degree B-spline surface using MATLAB R Spline Tool Box. Step 2: Form equations of motion 27 second-order ODEs. Step 3: Obtain initial conditions Initial conditions must satisfy no-slip and holonomic constraints. 3 actuated variables can be chosen arbitrarily. Rest 24 obtained using inverse kinematics equations. Step 4: Solve ODEs numerically ODE solver in MATLAB R used to obtain evolution of all generalised coordinates. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

73 NUMERICAL SIMULATIONS RESULTS C 3 φ 3 Top Platform θ 2 δ 2 P C 2 C 1 θ 3 θ 1 δ Center of 2.1 Top Platform {} O Figure 39: WMR G 2 G 3 G 3 DOF "Non holonomic" Joint Schematic of the 3 DOF Figure 4: B-spline surface with C 3 continuity Synthetic elevation data Uneven terrain generated using MATLAB R Spline Tool Box ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

74 NUMERICAL SIMULATIONS RESULTS Mass of platform = 1 kg, Mass of each wheel = 1 Kg. Maximum allowable deflection of δ i is π/4 under self-weight k si = N-m/rad and k di =.57 N-m-s/rad. Initial conditions: u 1 = 1.57, v 1 = 3π 2, u g 1 = 5.8 m, v g1 = m, ψ 1 = 3.142, u 2 = , v 2 = 3π 2, u g2 = 4 m, v g2 = 1.5 m, ψ 2 = , u 3 = , v 3 = 3π 2, u g 3 = m, v g3 = m, ψ 3 = , θ 1 =, θ 2 =, θ 3 =, δ 1 =, δ 2 =, ϕ 3 =, z c = m, α =.127, and β =.133 All angles in radians. All the initial values of the first derivatives are chosen to be. Inputs: τ 1 =.35 N-m, τ 2 =.5 N-m, and τ 3 =.1t N-m. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

75 NUMERICAL SIMULATIONS RESULTS 1 9 Wheel 3 Wheel ground contact point Center of wheel Center of platform.8 δ δ 2 7 Wheel 2 Wheel δ 1,δ 2 (radian) Figure 41: Locus of wheel-ground contact point, wheel centre and platform centre time (sec) Figure 42: Variation of δ 1 and δ 2 Uneven terrain δ 1 and δ 2 varies automatically to avoid slip. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

76 NUMERICAL SIMULATIONS RESULTS Error in holonomic constraints (m) x 1 6 Constraint12 Constraint13 Constraint time (sec) Figure 43: Satisfaction of holonomic constraints in dynamics Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) Slip velocity at wheel ground contact point (m/s) 1 x 1 7 vx vy x 1 7 time (sec) vx vy x 1 7 time (sec) 1 vx vy time (sec) Figure 44: Slip velocities at wheel-ground contact points for three wheels All constraints are satisfied at least up to 1 7 m. Three-wheeled mobile robot can traverses uneven terrain without slip!! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

77 STABILITY ANALYSIS Uneven terrain Loss of vehicle stability due to tip-over or roll over. Tip-over Vehicle undergoes rotation resulting in reduction in number of vehicle-ground contact points. Mobility is lost and, if rotational motion is not arrested, vehicle overturns. Need a measure of stability to warn operator. Placement of centre of mass, speed, acceleration, external forces/moments and nature of terrain determine tip-over or stability. Various measures of stability Force-angle stability measure (Papadopoulos and Rey, 1996) used. Investigate stability of the earlier studied three-wheeled mobile robots with torus-shaped wheels for different conditions. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

78 FORCE-ANGLE STABILITY MEASURE I 2 Y p c θ 2 Centre of mass subjected to a net force f r. p 1 X I 1 θ 1 fr f r makes angle θ 1 and θ 2 with tip-over axis normals I 1 and I 2. Force-angle stability measure ξ = min{θ 1,θ 2 } f r Figure 45: measure Planar force-angle stability ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

79 {} X Z FORCE-ANGLE STABILITY MEASURE FOR WMR Centre of Mass p c p p 1 Y 2 a 1 fr Wheel 1 nr θ 2 f* Wheel 2 Figure 46: Force-angle stability measure for WMR 2 I 2 a 3 p 3 a 2 Wheel 3 Wheel-ground contact points p i,= (u gi,v gi,z i ) T, i = 1,2,3 known. Location of centre of mass p c = (x c,y c,z c ) T known in {}. Line joining wheel-ground contact points a i, i = 1,2,3 are tip-over axis. Component of net resultant force is f 2 for tip-over axis a 2 (see next slide). Angle θ 2 for tip-over axis a 2. Likewise find θ 1 for a 1 and θ 3 for a 3. If any θ i =, WMR can tip-over a i. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

80 FORCE-ANGLE STABILITY MEASURE FOR WMR Net resultant force at centre of mass: f r = f gravity + f dist f inertial Force due to f gravity and inertial f inertial obtained from dynamic simulation. f rmdist External disturbance. Net resultant moment at centre of mass: n r = n gravity + n dist n inertial Interested in component f i and n i about tip-over axis a i. Combine f i and n i to get a net resultant force fi = f i + I i n i, I i is the tip-over axis normal. I i Angle for stability measure, θ i, angle between fi and unit vector along I i. Sign of θ i determines if the net resultant force in inside the support polygon or not. Overall force-angle stability measure: ξ i = min{θ 1,θ 2,θ 3 } f r ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, 21 8 / 88

81 NUMERICAL SIMULATION RESULTS Geometry, mass and inertial parameters same as in dynamic analysis Mass of platform = 1 kg and Mass of each wheel = 1 kg. Spring constant = Nm/rad and damping =.57 Nms/rad. Various terrains: Curved path on flat terrain, two rear wheels on two different planes and uneven terrains. For each chosen paths and/or input torques, compute at every instant of time t Net resultant force f r at centre of mass. Net resultant moment n r at centre of mass. Compute tip-over axis a i and normal I i for i = 1,2,3 Compute f i Compute force-angle stability measure ξ i. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

82 NUMERICAL SIMULATION RESULTS FLAT PLANE Figure 47: Curve path on a plane Figure 48: Stability margin for curve path on a plane Actuator torques (N-m): τ 1 =.5, τ 2 =.75, τ 3 =.4t. As the WMR turns, stability margin reduces. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

83 NUMERICAL SIMULATION RESULTS INCLINED PLANE 3 1 wheel 1 2 wheel 2 3 wheel stability margin 2 slip at wheel 1, wheel 2, wheel 3 Holonomic constraints Z in meters Y in meters X in meters Figure 49: Path traced by WMR on incline plane Figure 5: Time in seconds Stability margin for path on an inclined plane Input torques (N-m): τ 1 = 2.4, τ 2 = 4, τ 3 =.8t. Initially least stability about axis 2 As the WMR turns, tip-over starts shifting from axis 2 to axis 1 Stability increases. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

84 NUMERICAL SIMULATION RESULTS UNEVEN TERRAIN 1 Wheel 1 2 Wheel 2 3 Wheel Stability margin 1 Stability margin 2 Z 1 in meters Z 2 in meters Z 3 in meters Z in meters X in meters Y in meters Figure 51: WMR climbing obstacle on a straight path Time in seconds Figure 52: Stability margin along straight path Input torques (N-m): τ 1 = 4, τ 2 = 4., τ 3 =.. WMR is able to negotiate obstacle on uneven terrain without tip-over. ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88

85 Zc in meters 1 NUMERICAL SIMULATION RESULTS UNEVEN TERRAIN Yc in meters Xc in meters wheel 1 2 wheel 2 3 wheel 3 4 unstable Figure 53: Path traced by WMR on an uneven terrain Figure 54: Stability margin 2 Stability margin 1 Z 3 in meters Slip v1x in m/s Position of front wheel (v3g) in meters Stability margin on uneven terrain Input torques (N-m): τ 1 = 4, τ 2 = 4., τ 3 =.. ξ reduces while climbing second peak, tip-over occurs about axis 1. Simulation strictly not valid after the vertical line Wheel slip increases to more than 1 3 m! ASHITAVA GHOSAL (IISC) ROBOTICS: ADVANCED CONCEPTS & ANALYSIS NPTEL, / 88