Forward Kinematics and Configurations Kris Hauser I400/B659: Intelligent Robotics Spring 2014 1

Articulated Robot Robot: usually a rigid articulated structure Geometric CAD models, relative to reference frames A configuration specifies the placement of those frames (forward kinematics) q1 q2

Forward Kinematics Given: A kinematic reference frame of the robot Joint angles q1,…,qn Find rigid frames T1,…,Tn relative to T0 A frame T=(R,t) consists of a rotation R and a translation t so that T·x = R·x + t Make notation easy: use homogeneous coordinates Transformation composition goes from right to left: T1·T2 indicates the transformation T2 first, then T1

Kinematic Model of Articulated Robots: Reference Frame T2ref T3ref L2 T4ref T0 L3 T1ref L1 L0

Rotating the first joint T1(q1) = T1ref·R(q1) q1 T1(q1) T0 T1ref L0

Where is the second joint? T2(q1) ? T2ref T0 q1

Where is the second joint? T2ref T0 q1 T2parent(q1) = T1(q1) ·(T1ref)-1·T2ref

After rotating joint 2 q2 T2R T0 q1 T2(q1,q2) = T1(q1) ·(T1ref)-1·T2ref·R(q2)

After rotating joint 2 q2 T2R T0 q1 Denote T2->1ref = (T1ref)-1·T2ref (frame relative to parent) T2(q1,q2) = T1(q1) ·T2->1ref·R(q2)

General Formula Denote 𝑇 𝑖→𝑖−1 𝑟𝑒𝑓 = 𝑇 𝑖−1 𝑟𝑒𝑓 −1 𝑇 𝑖 𝑟𝑒𝑓 (ref frame relative to parent) 𝑇 𝑘 𝑞 1 ,…, 𝑞 𝑘 = 𝑖=1 𝑘 𝑇 𝑖→𝑖−1 𝑟𝑒𝑓 𝑅( 𝑞 𝑖 ) T2(q1,q2) T3(q1,..,q3) L2 T4(q1,…,q4) T0 L3 T1(q1) L1 L0

Generalization to tree structures Topological sort: p[k] = parent of link k Denote 𝑇 𝑖→𝑝[𝑖] 𝑟𝑒𝑓 = 𝑇 𝑝 𝑖 𝑟𝑒𝑓 −1 𝑇 𝑖 𝑟𝑒𝑓 (frame i relative to parent) Let A(i) be the list of ancestors of i (sorted from root to i) 𝑇 𝑘 𝑞 1 ,…, 𝑞 𝑘 = 𝑖∈𝐴(𝑘) 𝑇 𝑖→𝑝[𝑖] 𝑟𝑒𝑓 𝑅( 𝑞 𝑖 )

To 3D… Much the same, except joint axis must be defined (relative to parent) Angle-axis parameterization

Generalizations Prismatic joints Ball joints Cylindrical joints Spirals Free-floating bases From LaValle, Planning Algorithms

Configuration Space A robot configuration is a specification of the positions of all robot frames relative to a fixed coordinate system Usually a configuration is expressed as a “vector” of parameters q=(q1,…,qn) q1 q2 q3 qn

Rigid Robot 3-parameter representation: q = (x,y,q) workspace robot reference direction q y reference point x 3-parameter representation: q = (x,y,q) In a 3-D workspace q would be of the form (x,y,z,a,b,g)

Articulated Robot q1 q2 q = (q1,q2,…,q10)

Protein

Configuration Space q y x Space of all its possible configurations But the topology of this space is in general not that of a Cartesian space q q’ 3-D cylinder embedded in 4-D space y q robot x x y q 2p S1 R2S1

Configuration Space C = S1 x S1 Space of all its possible configurations But the topology of this space is in general not that of a Cartesian space C = S1 x S1

Configuration Space C = S1xS1 Space of all its possible configurations But the topology of this space is in general not that of a Cartesian space C = S1xS1

Configuration Space C = S1xS1 Space of all its possible configurations But the topology of this space is in general not that of a Cartesian space C = S1xS1

Some Important Topological Spaces R: real number line Rn: N-dimensional Cartesian space S1: boundary of circle in 2D S2: surface of sphere in 3D SO(2), SO(3): set of 2D, 3D orientations (special orthogonal group) SE(2), SE(3): set of rigid 2D, 3D translations and rotations (special Euclidean group) Cartesian product A x B, power notation An = A x A … x A Homeomorphism ~ denotes topological equivalence Continuous mapping with continuous inverse (bijective) Cube ~ S2 SO(2) ~ S1 SE(3) ~ SO(3) x R3

What is its topology? q1 q2 (S1)7xI3 (I: Interval of reals)

Notion of a (Geometric) Path q 1 3 n 4 2 t(s) A path in C is a piece of continuous curve connecting two configurations q and q’: t : s  [0,1]  t (s)  C

Examples A straight line segment linearly interpolating between a and b t(s) = (1-s) a + s b What about interpolating orientations? A polynomial with coeffients c0,…,cn t(s) = c0 + c1s + … + cnsn Piecewise polynomials Piecewise linear Splines (B-spline, hermite splines are popular) Can be an arbitrary curve Only limited by your imagination and representation capabilities

Notion of Trajectory vs. Path q 1 3 n 4 2 t(t) A trajectory is a path parameterized by time: t : t  [0,T]  t (t)  C

Translating & Rotating Rigid Robot in 2-D Workspace x y q reference point robot reference direction workspace q x y 2p configuration space What is the placement of the robot in the workspace at configuration (0,0,0)?

Translating & Rotating Rigid Robot in 2-D Workspace x y q reference point robot reference direction workspace q x y 2p configuration space What is the placement of the robot in the workspace at configuration (0,0,0)? 29

Translating & Rotating Rigid Robot in 2-D Workspace x y q reference point robot reference direction workspace q x y 2p configuration space What is this path in the workspace? What would be the path in configuration space corresponding to a full rotation of the robot about point P? P

Klamp’t Python API world = WorldModel() world.readFile([some file]) robot = world.robot(0) [if the world has only one robot] A robot’s configuration is a list of numbers. A RobotModel stores a current configuration robot.getConfig() robot.setConfig(q) automatically performs forward kinematics It does not necessarily transform like a vector! Robot-specific interpolation function: robot.interpolate(a,b,u) A robot’s frames are given as a list of RobotModelLink’s link = robot.getLink([index or name]) (R,t) = link.getTransform() gets the current transformation of the link (last set by setConfig) 3D rigid transform utilities in se3.py