ME/ECE 439 2007 Professor N. J. Ferrier
Forward Kinematics
Professor Nicola FerrierME Room 2246, 265-8793
ME/ECE 439 2007 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 439 2007 Professor N. J. Ferrier
Industrial Robot
sequence of rigid bodies (links) connected by
means of articulations
(joints)
ME/ECE 439 2007 Professor N. J. Ferrier
Robot Basics: Modeling
• Kinematics:– Relationship between
the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration
ME/ECE 439 2007 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 439 2007 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 439 2007 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 439 2007 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 qi (angles/displacements) and link geometry (link lengths and relative orientations)
Pe = f(q1,q2,,qn ;l1,ln,1,n)
ME/ECE 439 2007 Professor N. J. Ferrier
Pose of a Rigid Body
• Pose = Position + Orientation• Physical space, E3, 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 R3.– A point, P, is assigned a 3-vector:
AP = (x,y,z) where A denotes the frame of reference
ME/ECE 439 2007 Professor N. J. Ferrier
A BX
X
Y
Y
Z
Z
AP = (x,y,z)
BP = (x,y,z)
P
ME/ECE 439 2007 Professor N. J. Ferrier
Pose of a Rigid Body
• Pose = Position + Orientation
How do we do this?
ME/ECE 439 2007 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
XA
YA
ZA
ME/ECE 439 2007 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
XA
YA
ZA
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
• OXYZ & OUVW have coincident origins at O– OUVW is fixed to the object
– OXYZ has unit vectors in the directions of the three axes ix, jy,and kz
– OUVW has unit vectors in the directions of the three axes iu, jv,and kw
• Point P can be expressed in either frame:
ME/ECE 439 2007 Professor N. J. Ferrier
OXU
V
Y
W
ZAP = (x,y,z)
BP = (u,v,w)
P
ME/ECE 439 2007 Professor N. J. Ferrier
OXU
V
Y
W
ZAP = (x,y,z)
P
BP = (u,v,w)
ME/ECE 439 2007 Professor N. J. Ferrier
OXU
V
Y
W
ZAP = (x,y,z)
P
BP = (u,v,w)
ME/ECE 439 2007 Professor N. J. Ferrier
OXU
V
Y
W
ZAP = (x,y,z)BP = (u,v,w)
P
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
1
X axis expressed wrt Ouvw
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
1
Y axis expressed wrt Ouvw
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
1
Z axis expressed wrt Ouvw
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
Z axis expressed wrt Ouvw
X axis expressed wrt Ouvw
Y axis expressed wrt Ouvw
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
1
U axis expressed wrt Oxyz
ME/ECE 439 2007 Professor N. J. Ferrier
Rotation Matrices
U axis expressed wrt Oxyz
V axis expressed wrt Oxyz
W axis expressed wrt Oxyz
ME/ECE 439 2007 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 439 2007 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 439 2007 Professor N. J. Ferrier
Elementary Rotations: X axis
X Y
Z
ME/ECE 439 2007 Professor N. J. Ferrier
Elementary Rotations: X axis
X Y
Z
ME/ECE 439 2007 Professor N. J. Ferrier
Elementary Rotations: Y axis
X Y
Z
ME/ECE 439 2007 Professor N. J. Ferrier
Elementary Rotations: Z-axis
X Y
Z
ME/ECE 439 2007 Professor N. J. Ferrier
Composition of Rotation Matrices
• Express P in 3 coincident rotated frames
ME/ECE 439 2007 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 439 2007 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 439 2007 Professor N. J. Ferrier
Rot[Z,90]Rot[Y,-90]
Rot[Z,90] Rot[Y,-90]
ME/ECE 439 2007 Professor N. J. Ferrier
Rot[z,90]Rot[y,-90] Rot[y,-90] Rot[z,90]
ME/ECE 439 2007 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 439 2007 Professor N. J. Ferrier
Decomposition of Rotation Matrices
• Euler Angles
• Roll, Pitch, and Yaw
ME/ECE 439 2007 Professor N. J. Ferrier
Decomposition of Rotation Matrices
• Angle Axis
ME/ECE 439 2007 Professor N. J. Ferrier
Decomposition of Rotation Matrices
• Angle Axis
• Elementary Rotations
ME/ECE 439 2007 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 439 2007 Professor N. J. Ferrier
Spatial Description of Body
• position of the origin with an orientation
AX Y
Z
B
ME/ECE 439 2007 Professor N. J. Ferrier
Homogeneous Coordinates
• Notational convenience
ME/ECE 439 2007 Professor N. J. Ferrier
Composition of Homogeneous Transformations
• Before:
• After
ME/ECE 439 2007 Professor N. J. Ferrier
Homogeneous Coordinates
• Inverse Transformation
ME/ECE 439 2007 Professor N. J. Ferrier
Homogeneous Coordinates
• Inverse Transformation
Orthogonal: no matrix inversion!