ME Robotics Dynamics of Robot Manipulators Purpose: This chapter introduces the dynamics of mechanisms. A robot can be treated as a set of linked rigid bodies. Each link body experiences the motion dynamics contributed by its own joint motor plus the cumulative effect of the other links that form a dynamic chain. This means that we must recursively accumulate the net dynamics by moving from one link to the next. This approach is referred to as the Newton-Euler recursive equations. The equation types are distinguished as Newton for force equations and as Euler for moment equations. Purpose: This chapter introduces the dynamics of mechanisms. A robot can be treated as a set of linked rigid bodies. Each link body experiences the motion dynamics contributed by its own joint motor plus the cumulative effect of the other links that form a dynamic chain. This means that we must recursively accumulate the net dynamics by moving from one link to the next. This approach is referred to as the Newton-Euler recursive equations. The equation types are distinguished as Newton for force equations and as Euler for moment equations.

ME Robotics In particular, you will 1.Review the fundamental force and moment equations for rigid bodies. 2.Determine that the moment of forces applied to a rigid body is the rate of change of angular momentum if taken about the body’s center of mass or about an inertial point. 3.Apply Newton-Euler recursive equations for the connected rigid links of a mechanism. 4.Understand that forward recursion is used to propagate motion through the links, while backward recursion is used to propagate forces and torques through the links. In particular, you will 1.Review the fundamental force and moment equations for rigid bodies. 2.Determine that the moment of forces applied to a rigid body is the rate of change of angular momentum if taken about the body’s center of mass or about an inertial point. 3.Apply Newton-Euler recursive equations for the connected rigid links of a mechanism. 4.Understand that forward recursion is used to propagate motion through the links, while backward recursion is used to propagate forces and torques through the links.

ME Robotics Review of fundamental equations Given a system of particles translating through space, each particle i being acted upon by external force F i, and each particle located relative to an inertial reference frame, the governing equations are F c = m a c (6.10) M = (6.11) M c = c (6.12) Given a system of particles translating through space, each particle i being acted upon by external force F i, and each particle located relative to an inertial reference frame, the governing equations are F c = m a c (6.10) M = (6.11) M c = c (6.12) What is an inertial frame?

ME Robotics Review of fundamental equations where m = total mass (sum over all mass particles) a c = acceleration of center of mass (cm) of all mass particles F c = sum of external forces applied to system of particles as if applied at cm H i = r i x m i v i = angular momentum of particle i (also called moment of momentum) H, H c = angular momentum summed over all particles, measured about inertial point, cm point, respectively M, M c = moment of all external forces applied to system of particles, measured about inertial point, cm point, respectively where m = total mass (sum over all mass particles) a c = acceleration of center of mass (cm) of all mass particles F c = sum of external forces applied to system of particles as if applied at cm H i = r i x m i v i = angular momentum of particle i (also called moment of momentum) H, H c = angular momentum summed over all particles, measured about inertial point, cm point, respectively M, M c = moment of all external forces applied to system of particles, measured about inertial point, cm point, respectively

ME Robotics Rigid bodies in general motion (translating and rotating) Z X Y x y z   The time rate of change of any vector V capable of being viewed in either XYZ or xyz is [ ] XYZ = [ ] xyz +  x V where  is the angular velocity of a secondary translating, rotating reference frame (xyz). The time rate of change of any vector V capable of being viewed in either XYZ or xyz is [ ] XYZ = [ ] xyz +  x V where  is the angular velocity of a secondary translating, rotating reference frame (xyz).

ME Robotics Rigid bodies in general motion (translating and rotating) Z X Y x y z   Another common form of the equation is: = r +  x V (6.13) and when applied to rate of change of angular momentum becomes M = = r +  x H (6.16) which is referred to as Euler’s equation. Another common form of the equation is: = r +  x V (6.13) and when applied to rate of change of angular momentum becomes M = = r +  x H (6.16) which is referred to as Euler’s equation.

ME Robotics Rotating rigid body By integrating the motion over the rigid body, we can express the angular momentum relative to the xyz axes as H = H x i + H y j + H z k = (J xx  x + J xy  y + J xz  z ) i + (J yx  x + J yy  y + J yz  z ) j + (J zx  x + J zy  y + J zz  z ) k (6.20) By integrating the motion over the rigid body, we can express the angular momentum relative to the xyz axes as H = H x i + H y j + H z k = (J xx  x + J xy  y + J xz  z ) i + (J yx  x + J yy  y + J yz  z ) j + (J zx  x + J zy  y + J zz  z ) k (6.20) or in matrix form H = J  where J = inertia matrix moments products

ME Robotics Rotating rigid body Taking the derivative of (6.20) and substituting into (6.16), also assuming the body axes to be aligned with the principal axes, we get Euler’s moment equations: M x = J xx x + (J zz - J yy )  y  z (6.29a) M y = J yy y + (J xx - J zz )  x  z (6.29b) M z = J zz z + (J yy - J xx )  x  y (6.29c) Taking the derivative of (6.20) and substituting into (6.16), also assuming the body axes to be aligned with the principal axes, we get Euler’s moment equations: M x = J xx x + (J zz - J yy )  y  z (6.29a) M y = J yy y + (J xx - J zz )  x  z (6.29b) M z = J zz z + (J yy - J xx )  x  y (6.29c) What are principal axes?

ME Robotics Acceleration relative to a non- inertial reference frame y x z X Y Z   P  R r

ME Robotics Acceleration relative to a non- inertial reference frame By taking two derivatives and applying (6.13) appropriately, the absolute acceleration of point P can be shown to be a = +  x  +  x (  x   x (6.31) where = acceleration of xyz origin  x  = tangential acceleration  x (  x  = centripetal acceleration = acceleration of P relative to xyz 2  x = Coriolis acceleration By taking two derivatives and applying (6.13) appropriately, the absolute acceleration of point P can be shown to be a = +  x  +  x (  x   x (6.31) where = acceleration of xyz origin  x  = tangential acceleration  x (  x  = centripetal acceleration = acceleration of P relative to xyz 2  x = Coriolis acceleration

ME Robotics Acceleration relative to a non- inertial reference frame For the special case of xyz fixed to rigid body and P a point in the body, and (6.31) reduces to a = +  x  +  x (  x  (6.32) If P at cm, then a c = +  x  c +  x (  x  c  (6.33) For the special case of xyz fixed to rigid body and P a point in the body, and (6.31) reduces to a = +  x  +  x (  x  (6.32) If P at cm, then a c = +  x  c +  x (  x  c  (6.33)

ME Robotics Recursive Newton-Euler Equations (forward recursion for motion) Use Craig/Red D-H form

ME Robotics Recursive Newton-Euler Equations If v i = i and  i is defined to be the angular velocity of the i th joint frame x i y i z i with respect to base coordinates, then where describes the velocity of x i+1, y i+1, z i+1 relative to an observer in frame x i, y i, z i. If v i = i and  i is defined to be the angular velocity of the i th joint frame x i y i z i with respect to base coordinates, then where describes the velocity of x i+1, y i+1, z i+1 relative to an observer in frame x i, y i, z i.

ME Robotics Recursive Newton-Euler Equations Likewise, the acceleration becomes Defining  i+1 to be the absolute angular velocity of the i+1 frame and  to be the angular velocity of the i+1 frame relative to the i th frame: Likewise, the acceleration becomes Defining  i+1 to be the absolute angular velocity of the i+1 frame and  to be the angular velocity of the i+1 frame relative to the i th frame:

ME Robotics Recursive Newton-Euler Equations Taking one more derivative for angular acceleration:

ME Robotics Recursive Newton-Euler Equations Now applying the DH coordinate representation for manipulators:

ME Robotics Recursive Newton-Euler Equations Using the previous equations, we can generate the angular motion recursive equations:

ME Robotics Recursive Newton-Euler Equations The linear velocity and acceleration equations use the D-H forms: where i+1 is the translational velocity of x i+1, y i+1, z i+1 relative to x i, y i, z i The linear velocity and acceleration equations use the D-H forms: where i+1 is the translational velocity of x i+1, y i+1, z i+1 relative to x i, y i, z i

ME Robotics Recursive Newton-Euler Equations Substituting (6.59) – (6.62), we get the velocity and acceleration recursion equations: Note that  i+1 =  i for translational link i+1.

ME Robotics Recursive Newton-Euler Equations (backward recursion for forces and torques) X o Y o Z o Joint i+1 Link i N i p i F i r i,  i * z  i z i+1 i  i. c i

ME Robotics Recursive Newton-Euler Equations (backward recursion for forces and torques) Link i n i f i f i+1 n Joint Forces/Torques

ME Robotics Recursive Newton-Euler Equations Define the terms: m i = mass of link i r i = position of link i cm with respect to base coordinates F i = total force exerted on link i N i = total moment " " " " * J i = inertia matrix of link i about its cm determined in the X o Y o Z o axes f i = force exerted on link i by link i-1 n i = moment " " " “ Define the terms: m i = mass of link i r i = position of link i cm with respect to base coordinates F i = total force exerted on link i N i = total moment " " " " * J i = inertia matrix of link i about its cm determined in the X o Y o Z o axes f i = force exerted on link i by link i-1 n i = moment " " " “

ME Robotics Recursive Newton-Euler Equations For each link we must apply the N-E equations: The gravitational acceleration and damping torques will be added to the equations of motion later. For each link we must apply the N-E equations: The gravitational acceleration and damping torques will be added to the equations of motion later.

ME Robotics Recursive Newton-Euler Equations Now i is easily calculated by knowing the acceleration of the origin of the i th frame attached to link i at joint i. We locate link i cm with respect to x i y i z i by c i such that r i = c i + p i. The velocity of the cm of link i is obviously Now i is easily calculated by knowing the acceleration of the origin of the i th frame attached to link i at joint i. We locate link i cm with respect to x i y i z i by c i such that r i = c i + p i. The velocity of the cm of link i is obviously

ME Robotics Recursive Newton-Euler Equations To determine F i and N i define f i = force exerted on link i by link i-1 n i = moment " " " “ Then F i = f i – f i+1 (6.71) and N i = n i – n i+1 + (p i - r i ) x f i - (p i+1 - r i ) x f i+1 (6.72) = n i – n i+1 - c i x F i – s i+1 x f i+1 To determine F i and N i define f i = force exerted on link i by link i-1 n i = moment " " " “ Then F i = f i – f i+1 (6.71) and N i = n i – n i+1 + (p i - r i ) x f i - (p i+1 - r i ) x f i+1 (6.72) = n i – n i+1 - c i x F i – s i+1 x f i+1

ME Robotics Recursive Newton-Euler Equations The previous equations can be placed in the backwards recursion form to work from the forces/moments exerted on the hand backwards to the joint torques necessary to react to these hand interactions and move the manipulator: f i = f i+1 + F i (6.74) n i = n i+1 + c i x F i + s i+1 x f i+1 + N i (6.75) The previous equations can be placed in the backwards recursion form to work from the forces/moments exerted on the hand backwards to the joint torques necessary to react to these hand interactions and move the manipulator: f i = f i+1 + F i (6.74) n i = n i+1 + c i x F i + s i+1 x f i+1 + N i (6.75)

ME Robotics Recursive Newton-Euler Equations The motor torque  i required at joint i is the sum of the joint torque n i resolved along the revolute axis plus the damping torque,  i = n i ˙ z i + b i i (revolute)(6.76a) where b i is the damping coefficient. For a translational joint  i = l i f i ˙ z i + b i i (translational)(6.77a) where l i is the torque arm for motor i. The motor torque  i required at joint i is the sum of the joint torque n i resolved along the revolute axis plus the damping torque,  i = n i ˙ z i + b i i (revolute)(6.76a) where b i is the damping coefficient. For a translational joint  i = l i f i ˙ z i + b i i (translational)(6.77a) where l i is the torque arm for motor i.

ME Robotics And what about gravity? The effect of gravity on each link is accounted for by applying a base acceleration equal to gravity to the base frame of the robot: o = g z o with z o vertical. o is applied to the base link in equations (6.65) and (6.66) for i = 0 and this serves to transmit the acceleration of gravity to each link by recursion.

ME Robotics There are two basic problems with the derivation so far. What are they? Problem 1 - J i in (6.68) when resolved into base coordinates is a function of manipulator configuration. To avoid this unnecessary complexity, we apply the equations at the cm of each link where J i is constant. Problem 2 – The recursive relations have not resolved the various vectors from one joint frame to the next. We must adjust the equations accordingly.

ME Robotics Do we use the full homogeneous transformation in the recursive equations? We resolve the free vectors by applying the rotational sub- matrix of the D-H transformations for each joint frame to the recursive vectors, using the Craig/Red D-H representation. Let us also use Tsai’s notation. joint frame i+1 relative to joint frame i: joint frame i relative to joint frame i+1: We resolve the free vectors by applying the rotational sub- matrix of the D-H transformations for each joint frame to the recursive vectors, using the Craig/Red D-H representation. Let us also use Tsai’s notation. joint frame i+1 relative to joint frame i: joint frame i relative to joint frame i+1:

ME Robotics Revised angular motion equations Do you notice anything about the form of the D-H rotational sub-matrix?

ME Robotics Revised linear motion equations

ME Robotics Revised force and torque equations

ME Robotics Dynamics summary The N-E equations are applied recursively to generate the forces and torques at each joint motor. We first apply forward recursion to get the motion state for each link. We then use this motion state to propagate the forces and torques in backward recursion to each joint. The rotational sub-matrix of the D-H transformations must be applied to resolve the vectors correctly into each link’s joint frame.