机器人学导论chapter6Manipulator DynamicsOutlineIntroduction运动学控制Why do we need to study the dynamics 4Given a set of force or torques applied the manipulator, try to calculate how the manipulator Given a desired trajectory, try to find out the desired torque inputs to cause this motion. E.g.: ControlContents of Dynamics:5RobotTwo methods for formulating dynamics model ：OutlineAcceleration of a Rigid Bodycalled linear acceleration /angular acceleration :Vectors of linear acceleration / angular acceleration can be described in different reference frame1.Linear acceleration of rigid body:9frame {A} and frame {B} have coincident origins, the velocity This form of the equation will be useful when deriving the corresponding acceleration equation.1.Linear acceleration of rigid body:10By differentiating1.Linear acceleration of rigid body:11origins are not coincident, we add one term which gives the origin of {B}:The above equation will be utilized to calculate thelinear acceleration of a manipulator (both revolute joint and prismatic joint). 2.Angular acceleration of rigid body: Consider that frame {B} rotate with respect to frame {A} with and frame {C} rotate with respect to frame {B} with , we can obtain:By differentiating:Applying equation (6.6):This equation will be utilized to calculate the angularacceleration of the links of a manipulator.Outline Mass DistributionMass Distributionexpressed in the following matrix form：whereMass moments of inertia Mass products of inertia Mass Distribution16As note, the inertia tensor is a functionof the location and orientation of thereference frame.{C}Where {C} is located at the center ofmass of the body.Mass Distributionwhereparallel-axis theorem:If the reference frame {C}(body frame) are selected such that theproducts of inertia being set to zero, the axes of this reference frameare called “principal axes（主轴）”, and the mass moment arenoted as “principal moments of inertia（主惯性矩）”.Outline1. Newton-Euler Dynamic Formulation 19rate of change of the linearmomentum is equal to theapplied forceLinear Momentum （动量） 1. Newton-Euler Dynamic Formulation 20Angular Momentum （角动量）Inertia Tensor1. Newton-Euler Dynamic Formulation where m is the mass of a rigid body, represent inertia tensor , F C is the external force on the center of gravity, N is the torque on the rigid body, v C represent the translational velocity , while ω is the angular velocity .2. Iterative (递推)Newton-Euler Dynamic Formulation 2.1 Compute velocities and accelerations Angular velocity from link to link:By differentiating:When joint i +1 is prismatic:2. Iterative (递推)Newton-Euler Dynamic Formulation The linear velocity of each link-frame origin:By differentiating: When joint i +1 is prismatic:The linear acceleration of the center of mass of each link:2. Iterative (递推)Newton-Euler Dynamic Formulation2. Iterative (递推)Newton-Euler Dynamic Formulation The torque-balance relationship for link i :2. Iterative (递推)Newton-Euler Dynamic Formulation 26Rearrange the force and torque equations:These equations are evaluated link by link, starting from linkn and working inward toward the base of the robot (inward force iterations ).2. Iterative (递推)Newton-Euler Dynamic Formulation As in the static case, the required joint torque are Found by taking the component of the torque Applied by one link on its neighbor: prismatic:2. Iterative (递推)Newton-Euler Dynamic Formulation 28Outward iterations: i: 0→n292. Iterative (递推)Newton-Euler Dynamic Formulation 2. Iterative (递推)Newton-Euler Dynamic Formulation Inward iterations: i: n →1移动关节转动关节2. Iterative (递推)Newton-Euler Dynamic Formulation32Newton-Euler Formulation of Manipulator Dynamics All mass exists as a point mass : The vectors that locate the Center of mass for each link:The inertia tensor for each link:33Newton-Euler Formulation of Manipulator Dynamics : There are no force acting on the end-effector:The base of the robot is not rotating:To include gravity force, we will use:34Newton-Euler Formulation of Manipulator Dynamics : The rotation between successive 35Newton-Euler Formulation of Manipulator Dynamics iteration for link 1:36Newton-Euler Formulation of Manipulator Dynamics iteration for37Newton-Euler Formulation of Manipulator Dynamics iteration for 38Newton-Euler Formulation of Manipulator Dynamics : The outward iteration for 39Newton-Euler Formulation of Manipulator Dynamics Example:Step 4: The outward iteration for link 2:40Newton-Euler Formulation of Manipulator Dynamics : The outward iteration for link 2:41Newton-Euler Formulation of Manipulator Dynamics : The inward iteration for 42Newton-Euler Formulation of Manipulator DynamicsNewton-Euler Formulation of Manipulator Dynamics 44Newton-Euler Formulation of Manipulator Dynamics torque: Written in Matrices Form:Newton-Euler Formulation of Manipulator Dynamics Example:有效惯量(effective inertial)：关节i 的加速度在关节i 上产生的惯性力Newton-Euler Formulation of Manipulator Dynamics 耦合惯量(coupled inertial)：关节i,j 的加速度在关节j ,i 上产生的惯性力Newton-Euler Formulation of Manipulator Dynamics 向心加速度(acceleration centripetal)系数关节i,j 的速度在关节j ,i 上产生的向心力Newton-Euler Formulation of Manipulator Dynamics 哥氏加速度(Coriolis accelaration)系数：关节j,k 的速度引起的在关节i 上产生的哥氏力(Coriolis force)Newton-Euler Formulation of Manipulator Dynamics 重力项(gravity)：关节i,j 处的重力OutlineLagrangian Formulation of Manipulator Dynamics 51Lagrangian Formulation of Manipulator Dynamics522-links manipulatorand Potential Energy Lagrangian Formulation of Manipulator Dynamics 53Example:2-links manipulatorKinetic Energy K 2 and Potential Energy P 2 of link 2:where Lagrangian Formulation of Manipulator Dynamics542-links manipulatorLagrangian Formulation of Manipulator Dynamics55OutlineFormulation Manipulator Dynamics in Cartesian Space The above dynamic equations is developed interms of the manipulator joint angles (jointspace).We could use the serial-link nature ofmechanism to advantage in deriving theequations.Sometimes, it might be desirable to express thedynamics with respect to Cartesian variablesform.57Formulation Manipulator Dynamics in Cartesian Space58 whereSummarySummarize steps to form Lagrangian Equation of n-link。
