MEM38 Applied Autonomous obots Winter obot Kinematics
Coordinate Transformations Motivation Ultimatel, we are interested in the motion of the robot with respect to a global or inertial navigation frame Motion of robot components (e.g. wheels, sensor heads, etc.) are measured relative to the robot frame What is required then is a means b which we can easil relate robot motions, the positions of targets, etc. from the robot s reference frame back to the inertial frame This is readil accomplished through coordinate transformations. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall
obot Configurations and Configuration Spaces Configuration Specification of position for all points on the robot Configuration Space Set of all possible configurations of the robot For a planar rigid bod: What s the configuration of this sstem?. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 3
obot Position as a Vector In general, obot position at t. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 4
Kinematics Degrees of Freedom The minimum number of parameters to define a configuration In 3-D, a rigid link has 6 DOF Constraints remove DOF evolute & Prismatic joints impose 5 constraints. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5
Manipulator Kinematics Coordinates End Effector Joint Coordinates Task Coordinates. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 6
elative vs. Absolute Joint Angles Joint angles can be defined relative to the previous link Joint angles can be defined absolutel with respect to the world coordinate frame. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 7
Position Kinematics The forward kinematics define the end-effector position for a given set of joint values The inverse kinematics determine the joint values for a given end-effector position (when such a solution eists). Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 8
Forward Kinematics via Trigonometr. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 9
Coordinate Frames & obot Positions Definition: coordinate frame A set n of orthonormal basis vectors spanning n For eample, i, When representing a point p, we need to specif a coordinate frame With respect to O : With respect to O : j, k 5 p 6 p.8 4. Note: v p and v p are invariant geometric entities But the representation is dependant upon choice of coordinate frame. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall
Coordinate Transformations A Coordinate Transformation relates the position vector of an point in coordinate frame to the same point in coordinate frame We are interested in igid Transformations which reflect the relative position and orientation of coordinate frame with respect to another Here i denotes the position vector for point as viewed from coordinate frame i, i j denotes a rotation matri which describes the rotation necessar to align the ais of coordinate frame i to j, and i t j the translation from the origin of frame i to j The rigid transformation necessar to align coordinate frame i with j has the opposite effect of translating points from frame j to frame i. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall
otation Matrices A rotation matri rotates position vectors in reference frame (F) to position vectors in F In two dimensions sin sin where θ corresponds to the relative difference in orientation of F with respect to F In this definition, transforms a position vector in F to how the corresponding position vector would appear in frame F.. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall
Coordinate Transformation: From Pure otations. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 3
Coordinate Transformation: From Pure otations Eample. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 4
. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5 Projecting the aes of from o onto the aes of frame o otation matrices as projections, sin sin
. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 6 Properties of rotation matrices Inverse rotations: Or, another interpretation uses odd/even properties: T sin sin sin sin T sin sin
MEM38: Fundamental obotics I Fall 7 Properties of rotation matrices Inverse of a rotation matri: The determinant of a rotation matri is alwas ± + if we onl use right-handed convention T sin sin det sin sin. Siegwart, I. Nourbakhsh
Properties of rotation matrices Summar: Columns (rows) of are mutuall orthogonal Each column (row) of is a unit vector T det The set of all n n matrices that have these properties are called the Special Orthogonal group of order n SOn. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 8
MEM38: Fundamental obotics I Fall 9 3D rotations 3 SO z z z z z z General 3D rotation: Special cases Basic rotation matrices sin sin, sin sin, z sin sin,. Siegwart, I. Nourbakhsh
MEM38: Fundamental obotics I Fall Now assume p is a fied point on the rigid object with fied coordinate frame O The point p can be represented in the frame O (p ) again b the projection onto the base frame otational transformations w v u p p w v u z z z z z z z wz z v z u wz v u wz v u z wz v u wz v u wz v u z p p p p. Siegwart, I. Nourbakhsh
MEM38: Fundamental obotics I Fall otating a vector Another interpretation of a rotation matri: otating a vector about an ais in a fied frame E: rotate v about b / v sin sin / /, v v. Siegwart, I. Nourbakhsh
Composition of otation Matrices Since Associativit 3 3 And 3, SO SO 3 Then, w/ respect to the current frame E: three frames O, O, O p p p p p p p p BUT EMBE: In general, members of SO(3) do not commute. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall
Compositions of rotations E: represents rotation about the current -ais b f followed b about the current z-ais, f z, f sinf sinf sin f sin f sin sinf f sin sinf sin sinf f. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 3
Pure Translations A pure translation corresponds to a transformation when =I. The equation for a rigid transformation then reduces to. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 4
Pure Translations: Eample. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5
General igid Transformations. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 6
MEM38: Fundamental obotics I Fall 7 igid motions igid motion is a combination of rotation and translation Defined b a rotation matri () and a displacement vector (d) the group of all rigid motions (d,) is known as the Special Euclidean group, SE(3) Consider three frames, O, O, and O and corresponding rotation matrices, and Let d be the vector from the origin o to o, d from o to o For a point p attached to o, we can represent this vector in frames o and o : 3 3 d SO 3 3 3 SE SO d d p d d p d p p d p p. Siegwart, I. Nourbakhsh
MEM38: Fundamental obotics I Fall 8 Homogeneous transforms We can represent rigid motions (rotations and translations) as matri multiplication Define: Now the point p can be represented in frame O : Where the P and P are: d H d H P H H P, p P p P. Siegwart, I. Nourbakhsh
Homogeneous transforms The matri multiplication H is known as a homogeneous transform and we note that H SE3 Inverse transforms: H T T d. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 9
MEM38: Fundamental obotics I Fall 3 Homogeneous transforms Basic transforms: Three pure translation, three pure rotation,,, c b a c z b a Trans Trans Trans,,, c s s c c s s c c s s c z ot ot ot. Siegwart, I. Nourbakhsh
Homogeneous epresentation epresentation of points & vectors Properties. Sum & differences of vectors are vectors. Sum of a vector and a point is a point 3. Difference between two points is a vector 4. Sum of two points == meaningless. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 3
Manipulator epresentation DH Parameters Jacques Denavit and ichard Hartenberg (955) standardizes the coordinate frames for spatial linkages A method for efficientl describing a kinematic chain. http://www.outube.com/watch?v=ra9tmgtln8. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 3
D-H steps. Siegwart, I. Nourbakhsh [SPONG notes] MEM38: Fundamental obotics I Fall 33
D-H Eamples Planar Elbow Manipulator [SPONG notes]. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 34
D-H Eamples SCAA Manipulator (pick & place) [SPONG notes]. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 35
Homogeneous Transformations C B A. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 36
Motion Control (wheeled robots) equirements for Motion Control Kinematic / dnamic model of the robot Model of the interaction between the wheel and the ground Definition of required motion -> speed control, position control Control law that satisfies the requirements Localization "Position" Global Map Cognition Environment Model Local Map Path Perception eal World Environment Motion Control. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 37
Introduction: Mobile obot Kinematics Aim Description of mechanical behavior of the robot for design and control Similar to robot manipulator kinematics However, mobile robots can move unbound with respect to its environment there is no direct wa to measure the robot s position Position must be integrated over time Leads to inaccuracies of the position (motion) estimate -> the number challenge in mobile robotics Understanding mobile robot motion starts with understanding wheel constraints placed on the robots mobilit. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 38
. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 39 Introduction: Kinematics Model Goal: establish the robot speed as a function of the wheel speeds, steering angles, steering speeds and the geometric parameters of the robot (configuration coordinates). forward kinematics Inverse kinematics wh not ),,,,, ( m m n f T i i i ),, ( f T m m n ),,, ( m n f I I s(t) v(t) -> not straight forward. Siegwart, I. Nourbakhsh
epresenting obot Position epresenting to robot within an arbitrar initial frame Initial frame: X I, Y Y I I obot frame: X, Y Y obot position: I T X Mapping between the two frames T I P X I sin sin Y I X Eample: obot aligned with Y I X. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall I. Siegwart, I. Nourbakhsh 4 Y
Eample. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 4
Forward Kinematic Models Given a differential-drive robot. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 4
Wheel Kinematic Constraints: Assumptions Y I Y Movement on a horizontal plane Point contact of the wheels Wheels not deformable r Pure rolling v = at contact point v No slipping, skidding or sliding No friction for rotation around contact point Steering aes orthogonal to the surface Wheels connected b rigid frame (chassis) P X X I. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 43
Mobile obot Workspace: Degrees of Freedom A vehicle s degree of freedom (DOF) is equivalent to its maneuverabilit But what is the degree of vehicle s freedom in its environment? Car eample Workspace how the vehicle is able to move between different configuration in its workspace? The robot s independentl achievable velocities. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 44
Mobile obot Workspace: Degrees of Freedom, Holonom DOF degrees of freedom: obots abilit to achieve various poses DDOF differentiable degrees of freedom: obots abilit to achieve various path Holonomic obots A holonomic kinematic constraint can be epressed a an eplicit function of position variables onl A non-holonomic constraint requires a different relationship, such as the derivative of a position variable Fied and steered standard wheels impose non-holonomic constraints. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 45
Mobile obot Workspace: Eamples of Holonomic obots. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 46
Mobile obot Kinematics: Non-Holonomic Sstems I, s =s ; s =s ; s L =s L but: = ; = s L s s s L, s s Non-holonomic sstems differential equations are not integrable to the final position. the measure of the traveled distance of each wheel is not sufficient to calculate the final position of the robot. One has also to know how this movement was eecuted as a function of time.. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 47 I
Non-Holonomic Sstems: Mathematical Interpretation A mobile robot is running along a trajector s(t). At ever instant of the movement its velocit v(t) is: v( t) s t sin t t ds d dsin Function v(t) is said to be integrable (holonomic) if there eists a trajector function s(t) that can be described b the values,, and onl. s s(,, ) I v(t) s(t) I This is the case if s s ; s s ; s s With s = s(,,) we get for ds ds s d Condition for integrable function s d s d. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 48
. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 49 Non-Holonomic Sstems: The Mobile obot Eample In the case of a mobile robot where and b comparing the equation above with we find Condition for an integrable (holonomic) function: the second (-sin=) and third (=) term in equation do not hold! sin d d ds d s d s d s ds ; sin ; s s s s s s s s s ; ;
Differential Steering: Forward Kinematics Given the robot geometr and wheel speeds, what is the robot s velocit? Let: r wheel radius l ale length right wheel speed left wheel speed obot Position We want. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5
Differential Steering: Forward Kinematics (cont.) Given the robot geometr and wheel speeds, what is the robot s velocit? Let: r wheel radius l ale length right wheel speed left wheel speed Goal Forward Velocit Angular Velocit. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5
Differential Steering Instantaneous Center of Curvature ICC. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 5
Instantaneous Center of otation Ackermann Steering Biccle. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall. Siegwart, I. Nourbakhsh 53
World Coordinates. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 54
Differential Steering: Inverse Kinematics. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 55
Differential Steering Benefits Simple construction Zero minimum turning radius Drawbacks Small error in wheel speeds translates to large position errors equires two drive motors Wheels-first is dnamicall unstable. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 56
The Idealized Knife-Edge Constraint Configuration (pose): Single contact point at C Velocit at point C Velocit constrained to be along the knife edge Lateral Velocit = Constrained velocit: Position is NOT constrained!. Siegwart, I. Nourbakhsh MEM38: Fundamental obotics I Fall 57