Robotics (Kinematics) Winter 1393 Bonab University
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1 Robotics () Winter 1393 Bonab University
2 : most basic study of how mechanical systems behave Introduction Need to understand the mechanical behavior for: Design Control Both: Manipulators, Mobile Robots Manipulator robots: more matured vs. Mobile robots who are following Mobile robot community asks similar questions, Example: Workspace Manipulator: range of possible positions achieved by its end effector relative to its fixture to the environment Mobile robot: range of possible poses achieved in its environment Controllability Manipulator: (manner) active engagement of motors used to move from pose to pose in the workspace Mobile robot: defines possible paths and trajectories in its workspace 2
3 : how mechanical systems behave Introduction Dynamics: Manipulator: places additional constraints on workspace and trajectory due to mass and force considerations Mobile robot: limited by dynamics; for instance, a high center of gravity limits the practical turning radius of a fast, car-like robot What is the main difference? Position estimation: Manipulator: one end fixed understanding kinematics of the robot & measuring the position of all intermediate joints always computable by current sensor data Mobile robot: can wholly move no direct way to measure position instantaneously Instead: integrate the motion of the robot over time (Add to this the inaccuracies of estimation due to slippage) (precise) extremely challenging understanding the motions of a robot: 1. Each wheel s contribution (enabling to move) 2. Also imposing a constraint (e.g. refusing to skid laterally) 3
4 Mobile Robot Introduction This chapter overview: 1. Notation: allowing expression of robot motion in a global reference frame as well as the robot s local reference frame. 2. Demonstrate construction of simple forward kinematic models of motion (how robot as a whole moves as a function of Its geometry Individual wheel behavior 3. Formally describe the kinematic constraints of individual wheels combine these kinematic constraints to express the whole robot s kinematic constraints 4. With these tools: evaluate the paths and trajectories that define the robot s maneuverability 4
5 Mobile Robot Benefits / Usage Predict the movement Wheel Odometry (use of data from motion sensor to estimate position change over time) Find the distribution of speed and steering Design controller & path planner 5
6 Kinematic Models and Constraints Deriving a model for the whole robot s motion: a bottom-up process: Each individual wheel contributes: motion At the same time, imposes constraints on robot motion Wheels are tied together (chassis geometry) their constraints combine form: constraints on the overall motion of the robot chassis Forces and constraints of each wheel must be expressed with respect to a clear and consistent reference frame Also needed: a clear mapping between global and local frames of reference Representing robot position: Assumption: robot = a rigid body on wheels, operating on a horizontal plane The total dimensionality of this robot chassis on the plane =3 2: for position in the plane 1: for orientation along the vertical axis 6
7 Representing robot position Of course, there are additional DoF & flexibility (wheel axles, wheel steering joints, and wheel castor joints), but by robot chassis = only to the rigid body of the robot, ignoring the joints and DoF internal to the robot and its wheels Global reference frame of the plane v.s. local reference frame of the robot? Arbitrary: Origin (O) Arbitrary: X I, Y I (Inertial basis) Choose a point P on the robot chassis as its position reference point X R, Y R defines 2 axes relative to P on the robot chassis (robot s local reference) Pose of a Robot: 7 y O x
8 Rotation Matrix connects the frames of reference ( ξ I, ξ R ) ξ R =Known ξ I =? Pose change is known in Local frame global? Δθ ΔX I = ΔX R cos θ - ΔY R sin θ ΔY I = ΔX R sin θ + ΔY R cos θ Δθ I = Δθ R ξ I = ΔX I Δt ΔY I Δt Δθ I Δt = cos θ sin θ 0 sin θ cos θ ΔX R Δt ΔY R Δt Δθ R Δt ΔY I ξ I = R -1 (θ) ξ R 8 ΔX I
9 Describing robot motion in terms of component motions First: ξ I =Known ξ R =? (what is the command in robot s language) It s necessary: to map motion along the axes of the global reference frame to motion along the axes of the robot s local reference frame Of course, the mapping is a function of the current pose of the robot Mapping needs : orthogonal rotation matrix: R(θ) ξ I The operation is denoted: Computation of this operation depends on the value of θ The mobile robot aligned with a global axis 9
10 Describing robot motion in terms of component motions For example: Θ = 90 Given some velocity in the global-ref: Robot experiences what velocities? How about a bit afterwards? Forward : How does the robot move, given its geometry and the speeds of its wheels? P centered between the 2 wheels Diff. drive robot has two wheels, each with diameter r Each wheel is a distance l from P Spinning speed of each wheel, φ 1, φ 2 Forward kinematic predicts robot s overall speed in the global reference: ξ I = R -1 (θ) ξ R 10
11 Forward How to find f? 11
12 Forward How to find f? First: compute the contribution of each of the two wheels in the local reference: ξ R Our example of diff. drive robot: Contribution 1 wheel s spinning speed to the translation speed at P in the direction of +X R (P is half way) The other wheel stationary x r1 = 1 2 r φ 1 In the same way, wheel-2: x r2 = 1 2 r φ 2 In a diff. drive these 2 components can be simply added to form X R, of ξ R Assume two wheels spinning with the same speed in opposite directions X R =? Y R is even simpler to calculate, neither wheel can contribute to sideways Y R = 0 12
13 Forward Finally, we have to compute: θ R, rotational component of ξ R What is contribution of each wheel? Add them Consider right wheel (wheel-1) moves forward Robot rotates CCW Pivoting around Wheel-2 rotation velocity ω 1 ω 1 = r φ 1 2l The same calculation for Wheel-2, but its forward movement CW ω 2 = - r φ 2 2l Combining all above: cos θ sin θ 0 sin θ cos θ Example: Robot position: θ=90 r=1, l=1 Wheels are unevenly engaged: φ 1 =4, φ 2 =2 interpretation 2l 13
14 Wheel kinematic constraints Previous approach: Provided motion of a robot given its component wheel However, we wish to determine the space of possible motions for each robot chassis design must go further: describing the constraints on robot motion imposed by each wheel Simplifying Assumptions: Wheels remain vertical Single point of contact to ground No sliding at this single point of contact Under these assumptions: 2 constraints for every wheel type 1. rolling contact the wheel must roll when motion takes place in the appropriate direction. 2. No lateral slippage the wheel must not slide orthogonal to the wheel plane. 14
15 Wheel kinematic constraints: 1-Fixed standard wheel No vertical axis of rotation for steering angle to the chassis = fixed it is limited to motion back and forth Along the wheel plane Rotation around its contact point with the ground plane Wheel = A Position: polar coordinates by distance l & angle α Angle of wheel plane relative to the chassis: β=fixed With radius= r, can spin over time: φ(t) rolling constraint all motion along the direction of the wheel plane must be accompanied by the appropriate amount of wheel spin pure rolling at the contact point: 15
16 Wheel kinematic constraints: 1-Fixed standard wheel Necessary to transform from I to R frame: because all other parameters in the equation: l, α, β are in terms of local reference frame In the same way (along the green line): +θ O R +x O R 16
17 Wheel kinematic constraints: 1-Fixed standard wheel What is the meaning of those equations? Example: Wheel is in a position that: 2 nd equation? Further assume that local & global frames Are aligned meaning: constrains the component of motion along X I to be zero and since X I & X R are parallel in this example, the wheel is constrained from sliding sideways, as expected 17
18 Wheel kinematic constraints: 2-Steered standard wheel Differs from the fixed standard wheel: only an additional DoF Equations are exactly the same 1 exception: β β(t) Constraints are identical to those of the fixed standard wheel because φ have a direct impact on the instantaneous motion constraints of a robot β does not, only by integrating over time that changes in steering angle can affect the mobility of a vehicle 18
19 Wheel kinematic constraints: 3-Castor wheel Able to steer around a vertical axis This axis of rotation does not pass through ground contact Like steered standard wheel, the castor wheel has 2 time-varying parameters: φ(t) β(t): steering angle and orientation of AB +θ O R Rolling constraint? Movement along X R ( X R ) contribution along wheel plane? AA Movement along Y R ( Y R )? BB Movement along θ R ( θ R )? CC -90 [AA BB CC] x X R Y R θ R =all contributions along the wheel plane=wheel must roll this amount 19
20 Wheel kinematic constraints: 3-Castor wheel AA: Cos((α+β)-90)=Cos(90-(α+β)) = Sin(α+β) BB: Cos(180-(α+β))= -Cos(α+β) CC: l x Cos(180-β) = -l Cos(β) +θ O R Rolling constraint as before (offset axis plays no role in motion aligned with the wheel plane) (α+β) 20
21 Wheel kinematic constraints: 3-Castor wheel (sliding constraint ) This wheel: significant impact on the sliding constraint (lateral force on the wheel occurs at point A) +θ O R Lateral movement being zero is wrong Instead it s more like a rolling constraint: Motions orthogonal to wheel plane must be balanced by equivalent and opposite castor steering 21 [AA BB CC ] x X R Y R θ R =all contributions perpendicular to the wheel plane=wheel must steer minus this amount
22 Wheel kinematic constraints: 3-Castor wheel First assume β (steering) is locked & Find the lateral motion at ground contact +θ O R AA : Cos(180-(α+β))= -Cos(α+β) (direction) BB : Cos(90-(180-(α+β))= Sin (180-(α+β)) = Sin(α+β) CC : (How much move will rotation cause at Contact point?): l Cos(β-90)=l Sin(β) (at A) l Sin(β)+d (at B) +θ O R Now unlocked steering should Compensate this lateral skid with arm length = d 180-(α+β) 22
23 Wheel kinematic constraints: 3-Castor wheel Last result is critical to the success of castor wheels because by setting the value of β o (t) any arbitrary lateral motion can be acceptable. In steered standard wheel, steering action movement chassis. In castor wheel, steering action -> moves chassis (because of the offset, d) Meaning: for any chassis motion there exists some value for spin speed φ(t) and steering speed β(t) such that the constraints are met So, a robot with only castor wheels can move with any velocity in the space of possible robot motions Example: five-castor wheel office chair Can push it by hand in any direction Similarly, if 2-motors for any wheel any trajectory = possible Wheel kinematics almost complex, but do not impose any real constraints on the kinematics of a robot chassis 23
24 Wheel kinematic constraints: 4-Swedish wheel No vertical axis of rotation, yet moves omnidirectionally by adding a DoF to the fixed standard wheel (rollers) Attached to wheel perimeter with antiparallel axes to main axis Angle = γ (between roller axes and the Wheel plane) Usually, 0 / 45 deg Since each axis can spin CW/CCW combine any vector Along Axis-1 with any vector along axis-2 any direction Though Axes 1-2 not necessarily independent (except for 90-deg) Formulating constraint has some subtlety: The instantaneous constraint is due to orientation of the small rollers. Rollers spin axis has zero component of velocity at the contact point (moving in that direction without spinning the main axis & without sliding is not possible) 24
25 Wheel kinematic constraints: 4-Swedish wheel Motion constraint: Looks identical to the rolling constraint for the fixed standard wheel except that formula is modified by adding γ Therefore, effective direction along which the rolling constraint holds: Along this zero component rather than along the wheel plane Orthogonal to this direction the motion is not constrained because of the free rotation of rollers Behavior of this constraint and thereby the Swedish wheel changes dramatically as γ varies Example: γ = 0 rolling constraint = like fixed standard wheel No sliding constraint γ = 90 No sliding constraint No benefit (in terms of lateral freedom of motion) 25
26 Wheel kinematic constraints: 4-Swedish wheel 26
27 Wheel kinematic constraints: 5-Spherical wheel Ball or spherical wheel No direct constraints on motion No principal axis of rotation No appropriate rolling or sliding constraints Clearly omnidirectional as previous 2-types Describes the roll rate of the ball in the direction of motion V A By definition the wheel rotation orthogonal to this direction = 0: Special case = moving along YR Equations for the spherical wheel are exactly the same as for the fixed standard wheel However, the interpretation is different: The omnidirectional spherical wheel can have any arbitrary direction of movement, where the motion direction given by β is a free variable 27
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