ME/ECE Professor N. J. Ferrier Forward Kinematics Professor Nicola Ferrier ME Room 2246,

ME/ECE Professor N. J. Ferrier Forward Kinematics Modeling assumptions Review: –Spatial Coordinates Pose = Position + Orientation –Rotation Matrices –Homogeneous Coordinates Frame Assignment –Denavit Hartenberg Parameters Robot Kinematics –End-effector Position, –Velocity, & –Acceleration Today Next Lecture

ME/ECE Professor N. J. Ferrier Industrial Robot sequence of rigid bodies (links) connected by means of articulations (joints)

ME/ECE Professor N. J. Ferrier Robot Basics: Modeling Kinematics: –Relationship between the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration   

ME/ECE Professor N. J. Ferrier Modeling Robot Manipulators Open kinematic chain (in this course) One sequence of links connecting the two ends of the chain (Closed kinematic chains form a loop) Prismatic or revolute joints, each with a single degree of mobility Prismatic: translational motion between links Revolute: rotational motion between links Degrees of mobility (joints) vs. degrees of freedom (task) Positioning and orienting requires 6 DOF Redundant: degrees of mobility > degrees of freedom Workspace Portion of environment where the end-effector can access

ME/ECE Professor N. J. Ferrier Modeling Robot Manipulators Open kinematic chain –sequence of links with one end constrained to the base, the other to the end-effector Base End-effector

ME/ECE Professor N. J. Ferrier Modeling Robot Manipulators Motion is a composition of elementary motions Base End-effector Joint 1 Joint 2 Joint 3

ME/ECE Professor N. J. Ferrier Kinematic Modeling of Manipulators Composition of elementary motion of each link Use linear algebra + systematic approach Obtain an expression for the pose of the end-effector as a function of joint variables q i (angles/displacements) and link geometry (link lengths and relative orientations) P e = f(q 1,q 2,,q n ;l 1 ,l n, 1 , n )

ME/ECE Professor N. J. Ferrier Pose of a Rigid Body Pose = Position + Orientation Physical space, E 3, has no natural coordinates. In mathematical terms, a coordinate map is a homeomorphism (1-1, onto differentiable mapping with a differentiable inverse) of a subset of space to an open subset of R 3. –A point, P, is assigned a 3-vector: A P = (x,y,z) where A denotes the frame of reference

ME/ECE Professor N. J. Ferrier A B X X Y Y Z Z A P = (x,y,z) B P = (x,y,z) P

ME/ECE Professor N. J. Ferrier Pose of a Rigid Body Pose = Position + Orientation How do we do this?

ME/ECE Professor N. J. Ferrier Pose of a Rigid Body Pose = Position + Orientation Orientation of the rigid body –Attach a orthonormal FRAME to the body –Express the unit vectors of this frame with respect to the reference frame XAXA YAYA ZAZA

ME/ECE Professor N. J. Ferrier Pose of a Rigid Body Pose = Position + Orientation Orientation of the rigid body –Attach a orthonormal FRAME to the body –Express the unit vectors of this frame with respect to the reference frame XAXA YAYA ZAZA

ME/ECE Professor N. J. Ferrier Rotation Matrices O XYZ & O UVW have coincident origins at O –O UVW is fixed to the object –O XYZ has unit vectors in the directions of the three axes i x, j y,and k z –O UVW has unit vectors in the directions of the three axes i u, j v,and k w Point P can be expressed in either frame:

ME/ECE Professor N. J. Ferrier O X U V Y W Z A P = (x,y,z) B P = (u,v,w) P

ME/ECE Professor N. J. Ferrier O X U V Y W Z A P = (x,y,z) P B P = (u,v,w)

ME/ECE Professor N. J. Ferrier O X U V Y W Z A P = (x,y,z) P B P = (u,v,w)

ME/ECE Professor N. J. Ferrier O X U V Y W Z A P = (x,y,z) B P = (u,v,w) P

ME/ECE Professor N. J. Ferrier Rotation Matrices

ME/ECE Professor N. J. Ferrier Rotation Matrices 1 X axis expressed wrt O uvw

ME/ECE Professor N. J. Ferrier Rotation Matrices 1 Y axis expressed wrt O uvw

ME/ECE Professor N. J. Ferrier Rotation Matrices 1 Z axis expressed wrt O uvw

ME/ECE Professor N. J. Ferrier Rotation Matrices

ME/ECE Professor N. J. Ferrier Rotation Matrices Z axis expressed wrt O uvw X axis expressed wrt O uvw Y axis expressed wrt O uvw

ME/ECE Professor N. J. Ferrier Rotation Matrices 1 U axis expressed wrt O xyz

ME/ECE Professor N. J. Ferrier Rotation Matrices U axis expressed wrt O xyz V axis expressed wrt O xyz W axis expressed wrt O xyz

ME/ECE Professor N. J. Ferrier Properties of Rotation Matrices Column vectors are the unit vectors of the orthonormal frame –They are mutually orthogonal –They have unit length The inverse relationship is: –Row vectors are also orthogonal unit vectors

ME/ECE Professor N. J. Ferrier Properties of Rotation Matrices Rotation matrices are orthogonal The transpose is the inverse: For right-handed systems –Determinant = -1(Left handed) Eigenvectors of the matrix form the axis of rotation

ME/ECE Professor N. J. Ferrier Elementary Rotations: X axis X Y Z

ME/ECE Professor N. J. Ferrier Elementary Rotations: X axis X Y Z

ME/ECE Professor N. J. Ferrier Elementary Rotations: Y axis X Y Z

ME/ECE Professor N. J. Ferrier Elementary Rotations: Z-axis X Y Z

ME/ECE Professor N. J. Ferrier Composition of Rotation Matrices Express P in 3 coincident rotated frames

ME/ECE Professor N. J. Ferrier Composition of Rotation Matrices Recall for matrices AB  BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90]

ME/ECE Professor N. J. Ferrier Composition of Rotation Matrices Recall for matrices AB  BA (matrix multiplication is not commutative) Rot[Z,90] Rot[Y,-90]

ME/ECE Professor N. J. Ferrier Rot[Z,90] Rot[Y,-90] Rot[Z,90] Rot[Y,-90]

ME/ECE Professor N. J. Ferrier Rot[z,90]Rot[y,-90]  Rot[y,-90] Rot[z,90]

ME/ECE Professor N. J. Ferrier Decomposition of Rotation Matrices Rotation Matrices contain 9 elements Rotation matrices are orthogonal –(6 non-linear constraints) 3 parameters describe rotation Decomposition is not unique

ME/ECE Professor N. J. Ferrier Decomposition of Rotation Matrices Euler Angles Roll, Pitch, and Yaw

ME/ECE Professor N. J. Ferrier Decomposition of Rotation Matrices Angle Axis

ME/ECE Professor N. J. Ferrier Decomposition of Rotation Matrices Angle Axis Elementary Rotations

ME/ECE Professor N. J. Ferrier Pose of a Rigid Body Pose = Position + Orientation Ok. Now we know what to do about orientation…let’s get back to pose

ME/ECE Professor N. J. Ferrier Spatial Description of Body position of the origin with an orientation A X Y Z B

ME/ECE Professor N. J. Ferrier Homogeneous Coordinates Notational convenience

ME/ECE Professor N. J. Ferrier Composition of Homogeneous Transformations Before: After

ME/ECE Professor N. J. Ferrier Homogeneous Coordinates Inverse Transformation

ME/ECE Professor N. J. Ferrier Homogeneous Coordinates Inverse Transformation Orthogonal: no matrix inversion!