Forward Kinematics
Professor Nicola Ferrier
ME Room 2246, 265-8793
ferrier@engr.wisc.edu
ME/ECE 439 2007
Professor N. J. Ferrier
Forward Kinematics
• Modeling assumptions
• Review:
Today
– Spatial Coordinates
• Pose = Position + Orientation
– Rotation Matrices
– Homogeneous Coordinates
• Frame Assignment
– Denavit Hartenberg Parameters
• Robot Kinematics
– End-effector Position,
– Velocity, &
– Acceleration
ME/ECE 439 2007
Professor N. J. Ferrier
Next
Lecture
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
End-effector
Base
ME/ECE 439 2007
Professor N. J. Ferrier
Modeling Robot Manipulators
• Motion is a composition of elementary
motions
Joint 2
Joint 1
Joint 3
Base
ME/ECE 439 2007
End-effector
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
P
Z
BP
AP
X
A
= (x,y,z)
= (x,y,z)
Y
B
Y
X
ME/ECE 439 2007
Z
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
P
Z
AP
= (x,y,z)
BP = (u,v,w)
W
V
X
O
Y
U
ME/ECE 439 2007
Professor N. J. Ferrier
P
Z
AP
= (x,y,z)
BP = (u,v,w)
W
V
X
O
Y
U
ME/ECE 439 2007
Professor N. J. Ferrier
P
Z
AP
= (x,y,z)
BP = (u,v,w)
W
V
X
O
Y
U
ME/ECE 439 2007
Professor N. J. Ferrier
P
Z
AP
= (x,y,z)
BP = (u,v,w)
W
V
X
O
Y
U
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
X axis
expressed
wrt Ouvw
Y axis
expressed
wrt Ouvw
Z 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
ME/ECE 439 2007
V axis
expressed
wrt Oxyz
Professor N. J. Ferrier
W axis
expressed
wrt Oxyz
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
Z
X
ME/ECE 439 2007
Y
Professor N. J. Ferrier
Elementary Rotations: X axis
Z
X
ME/ECE 439 2007
Y
Professor N. J. Ferrier
Elementary Rotations: Y axis
Z
X
ME/ECE 439 2007
Y
Professor N. J. Ferrier
Elementary Rotations: Z-axis
Z
X
ME/ECE 439 2007
Y
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]
ME/ECE 439 2007
Rot[Y,-90]
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]
ME/ECE 439 2007
Rot[Y,-90]
Rot[Z,90]
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
Z
B
X
A
ME/ECE 439 2007
Y
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!
ME/ECE 439 2007
Professor N. J. Ferrier