lecture 10 forward kinematics Katie DC Feb. 25, 2020 Modern Robotics Ch. 4 Admin • Guest Lecture on Thursday 2/27 • Quiz 1 re-take is this week • HW3: PF Implementation due next week • HW5 due this week • I’ll be posting a course feedback form this week What is Forward Kinematics? • Kinematics: a branch of classical mechanics that describes motion of bodies without considering forces. AKA “the geometry of motion” • Forward kinematics: a specific problem in robotics. Given the individual state of each joint of the robot (in a local frame), what is the position of a given point on the robot in the global frame? Where is forward kinematics used? Forward Kinematics of a Simple Chain Assumptions • Our robot is a kinematic chain, made of rigid links and movable joints • No branches or loops (will discuss later) • All joints have one degree of freedom and are revolute or prismatic Review of Screw Motions We derived a representation for the screw axis: 휔 풮 = ∈ ℝ6 푣 where either • 휔 = 1 • where 푣 = −휔 × 푞 + ℎ휔, where 푞 is the point on the axis of the screw and ℎ is the pitch of the screw • 휔 = 0 and 푣 = 1 Screw Motions as Matrix Exponential The screw axis 풮푖 can be expressed in matrix form as: 휔 푣 풮 = 푖 ∈ 푠푒 3 푖 0 0 To express a screw motion given a screw axis, we use the matrix exponential: 푒 풮 휃 ∈ 푆퐸 3 Modeling Robot Joints as Screw Motions Case 1: Revolute Joint • 휔 = 1 • 푣 = −휔 × 푞 + ℎω • 푣 = −휔 × 푞 Modeling Robot Joints as Screw Motions Case 2: Prismatic Joint • 휔 = 0 • 푣 = 1 • Axis of movement defines 푣 Product of Exponentials Approach Let each joint have a configuration defined by 휃푖 Initialization steps: • Choose a fixed frame 푠 • Choose an end-effector (tool) frame attached to the robot 푏 • Put all joints in zero position • Let 푀 ∈ 푆퐸 3 be the configuration of 푏 in the 푠 frame when the robot is in the zero position Product of Exponentials Approach • Given zero position 푀 • For each joint , define the screw axis • For each motion of a joint, define the screw motion • These operations compose nicely through multiplication, giving us the Product of Exponentials (PoE) formula! 푇 휃 = 푒 풮1 휃1푒 풮2 휃2 ⋯ 푒 풮푛−1 휃푛−1푒 풮푛 휃푛푀 Visualizing the Product of Exponentials Example 1 Example 1: PoE Compute 푒 풮푖 휃푖 for each joint: 휔푖 휃푖 2 푒 풮푖 휃푖 = 푒 퐼휃푖 + 1 − cos 휃푖 휔푖 + 휃푖 − sin 휃푖 휔푖 푣푖 0 1 and compose with 푀: 푇 휃 = 푒 풮1 휃1푒 풮2 휃2푒 풮3 휃3푀 Example 2 Summary • Learned the basics of forward kinematics, which gives us a model for computing position and orientation of the end-effector • Uses the product of exponentials formula to define this transformation as composed matrix multiplications • Next time, we’ll learn about exponentials in the end-effector frame and modeling robots with the Universal Robot Description Format.