Modeling and Motion Planning for Mechanisms on a Non-Inertial Base

9 IEEE International Conference on Rootics and Automation Koe International Conference Center Koe, Japan, May 1-17, 9 Modeling and Motion Planning for Mechanisms on a Non-Inertial Base Pål J. From 1 Vincent Duindam Jan T. Gravdahl 1 Shankar Sastry 1 Department of Eng. Cyernetics, Norwegian Univ. of Science and Technology, Trondheim, Norway Department of EECS, University of California, 53 Cory Hall, Berkeley, CA 97-177, USA {from,tommy.gravdahl}@itk.ntnu.no v.duindam@ieee.org sastry@eecs.erkeley.edu Astract Rootic manipulators on ships and platforms suffer from large inertial forces due to the non-inertial motion of the ship or platform. When operating in high sea state, operation of such manipulators can e made more efficient and roust if these non-inertial effects are taken into account in the motion planning and control systems. Motivated y this application, we present a rigorous and singularity-free formulation of the dynamics of a rootic manipulator mounted on a non-inertial ase. We extend the classical dynamics equations for a serial manipulator to include the -DoF motion of the non-inertial ase. Then, we show two examples of a 1-DoF and a -DoF manipulator to illustrate how these non-inertial effects can e taken into account in the motion planning. q q 1 q 3 q Ψ Ψ I. INTRODUCTION The use of unmanned and autonomous vehicles operating in hostile environments has shown oth to e cost efficient and to protect humans from potentially dangerous situations. One such hostile environment is high sea state. We look into the case when a manipulator mounted on a ship or a platform is required to operate independently of the sea state. Large inertial forces may influence the manipulator and make the operation inaccurate, extremely energy demanding, or impossile due to torque limits. The inertial forces thus need to e taken into account in oth the path planning and control of the root. Ships and oil platforms are expected to ecome increasingly unmanned in the future and hence the need for continuously operating roots for surveillance, maintenance, and operation will grow [1, [. All these tasks ecome increasingly important in harsh environments such as high sea state. To e ale to continue operation under these conditions, a good understanding is needed of the effects of the inertial forces due to the motion of the ship or platform. We therefore develop the dynamic equations of the rootic manipulator including these effects. Research on several related topics can e found in literature. Love et al. [1 addressed the impact of wave generated disturances on the tracking control of a manipulator mounted on a ship ased on the classical Lagrangian approach. Repetitive learning control was used and performance was improved for purely periodic motions, ut no formal derivation of the dynamics equations was presented. The use of cale roots for loading and unloading cargo etween two ships has also een addressed y Kitarovic et al. [ and Oh et al. [3. In the Ampelmann project [, a Stewart platform is mounted on a ship and is used to compensate Fig. 1. Model setup for a four-link root attached to a non-inertial ase with coordinate frame Ψ. Frame Ψ denotes the inertial reference frame. for the motion of the ship y keeping the platform still with respect to the world frame. Leans et al. [5 give a cursory description of a telerootic shipoard handling system, and Kosuge et al. [, [7 addresses the control of roots floating on the water utilizing vehicle restoring forces. Other related research areas are macro/micro manipulators [, [9, underwater vehicle/manipulator systems [1 and spacecraft/manipulator systems [11. Most previous work deals with roots mounted on a free-floating ase. There is, however, an important difference etween modeling a root on a forced and a free-floating ase. A forced ase motion will add inertial forces to the dynamic equations that do not arise in free-floating case, such as spacecraft/manipulator systems and manipulators on small AUVs. Our approach differs from previous work in that the dynamic equations are derived for rigid multiody systems including oth Euclidean joints and generalized joints with configuration spaces different from R n. We follow the generalized Lagrangian approach presented in Duindam et al. [1, [13, which allows us to comine the Euclidean joints (the manipulator) and more general joints (the ase), i.e. joints that can e descried y the Lie group SE(3) or one of its ten sugroups. In our case, the transformation from the inertial frame to the ase frame (the platform) is represented as a free motion joint descried y the Lie group SE(3). We also show through the examples how the ase motion can e expressed as sugroups of SE(3), in our case SO(). For marine vessels in high sea state, very large inertial 97-1--79-5/9/$5. 9 IEEE 33

forces are added to the manipulator dynamics. To illustrate the effect of the inertial forces and how these appear in the root equations, we look into the prolem of finding the optimal trajectory in terms of actuator torque. There are many motivations for doing this. First of all the wear and tear on the manipulator is reduced and, for cooperative manipulation, the possiility of reaking an oject manipulated y two roots is also reduced. Secondly, the solution is more energy efficient as the inertial forces will, if possile, contriute to the desired motion instead of working against it. The final motivation is that a good understanding of the effects of the inertial forces on the dynamic equations is essential in tasks that require high accuracy and the need to compensate for these effects [1. We assume that the motion of the free moving ase is forced externally y forces unknown to us and that the pose, velocity, and acceleration of the ase relative to the inertial world are known for all times. This means we also assume that the motion of the root does not influence the motion of the ase. The motion of the ase will add inertial forces to the dynamics equations of the root and the pose of the ase will influence the gravitational forces acting on each link. Finally, we assume the root to e an ideal rigid frictionless mechanism with purely torque driven actuators. Given these assumptions, we consider the following two prolems: first, we derive the dynamic equations descriing the motion of the root under the influence of the non-inertial ase motion. Second, we consider the path planning prolem of finding the trajectory etween two given configurations for a given ase motion. Intuitively, the optimal solution is the trajectory for which the inertial forces help accelerate and decelerate the root as much as possile, such that little control torque is required. II. MULTIBODY DYNAMICS WITH A NON-INERTIAL BASE We extend the classical dynamics equations for a serial manipulator arm with 1-DoF joints to include the forced - DoF motion of the non-inertial ase. A. Manipulator kinematics on a non-inertial ase Consider the setup of Figure 1 descriing a general n-link root manipulator arm attached to a moving ase. Choose an inertial coordinate frame Ψ, a frame Ψ rigidly attached to the moving ase, and n frames Ψ i (not shown) attached to each link i at the center of mass with axes aligned with the principal directions of inertia. Finally, choose a vector q R n that descries the configuration of the n joints. Using standard notation [1, we can descrie the pose of each frame Ψ i relative to Ψ as a homogeneous transformation matrix g i SE(3) of the form [ Ri p g i i R (1) 1 with rotation matrix R i SO(3) and translation vector p i R 3. This pose can also e descried using the vector of joint coordinates q as g i g g i g g i (q) () The ase pose g and the joint positions q thus fully determine the configuration state of the root. In a similar way, the spatial velocity of each link can e expressed using twists [1: [ Vi v i V + Vi ( Ad g V + J i (q) ) (3) ω i where vi and ωi are the linear and angular velocities, respectively, of link i relative to the inertial frame, J i (q) R n is the geometric Jacoian of link i relative to Ψ, the adjoint is defined as Ad g : [ R ˆpR R R, and ˆp R 3 3 is the skew symmetric matrix such that ˆpx p x for all p,x R 3. The velocity state is thus fully determined given the twist V of the ase and the joint velocities. B. Manipulator dynamics on a non-inertial ase The previous section shows how the kinematics of the system can e naturally descried in terms of the (gloal) state variales g, q, V, and. We now derive the dynamics equations for the system in terms of these state variales. We first assume the ase to e free-moving under the influence of some prescried external wrench F, and then restrict the ase motion to e kinematically constrained. To derive the dynamics of the complete mechanism (including the -DoF etween Ψ and Ψ ), we follow the generalized Lagrangian method introduced y Duindam et al. [1, [13. This method gives the dynamics equations for a general mechanism descried y a set Q {Q i } of configuration states Q i (not necessarily Euclidean), a vector v of velocity states v i R ni, and several mappings that descrie the local Euclidean structure of the configuration states and their relation to the velocity states. More precisely, the neighorhood of every state Q i is locally descried y a set of Euclidean coordinates φ i R ni as Q i Q i ( Q i,φ i ) with Q i ( Q i,) Q i, and there exist differentiale matrices S i such that we can write v i S i (Q i,φ i ) φ i for every Q i. Given a mechanism with coordinates formulated in this generalized form, we can write its kinetic energy as U k (Q,v) 1 vt M(Q)v with M(Q) the inertia matrix in coordinates Q. The dynamics of this system then satisfy M(Q) v + C(Q,v)v τ () with τ the vector of gravitational, friction, and other external torques (collocated with v), and C(Q,v) the matrix descriing Coriolis and centrifugal forces and given y C ij (Q,v) : ( Mij S 1 kl 1 ) M jl φ k S 1 ki v l φ k k,l φ + ( ( S 1 Smj mi S ) kl) (5) ms S 1 sk φ s φ M v l j φ k,l,m,s More details and proofs can e found in references [1 and [13. To apply this general result to systems of the form of Figure 1, we write Q {g,q} as the set of configuration states, and v [ V as the vector of velocity states. 331

The local Euclidean structure for the state g is given y exponential coordinates [1, while the state q is gloally Euclidean of itself. Mathematically, we can express configurations (g,q) around a fixed state (ḡ, q) as g ḡ exp j (φ ) j () j1 q i q i + φ i i {1,...,n} (7) with j the standard asis elements of the Lie algera se(3). The corresponding matrices S i can e collected in one lockdiagonal matrix S given y [15 [( I 1 S(Q,φ) ad φ + 1 ad φ... ) I [ ˆp... ˆp 1...3 ˆp... R (+n) (+n) with ad p R for p R. This shows that the choice of coordinates (Q,v) has the required form. From expression (3) for the twist of each link in the mechanism, we can derive an expression for the total kinetic energy. Let I i R denote the constant positive-definite diagonal inertia tensor of link i expressed in Ψ i. The kinetic energy U k,i of link i then follows as U k,i 1 ( ) V i T i Ii Vi i 1 ( V + J i (q) ) T ( Ad T gi I i Ad gi V + J i (q) ) 1 [ (V ) [ T V T M i (q) 1 vt M i (q)v () with [ Ad T M i (q) : I gi i Ad gi Ad T g I i i Ad J gi i Ji T Ad T g i I i Ad gi Ji T Ad T g i I i Ad gi J i The total kinetic energy of the mechanism is given y the sum of the kinetic energies of the mechanism links and the non-inertial ase, that is, ( U k (q,v) 1 [I ) vt + M i (q) v (1) i1 inertia matrix M(q) with M(q) the inertia matrix of the total system. Note that neither U k (q,v) nor M(q) depend on the pose g and hence the choice of inertial reference frame Ψ. We can write () in lock-form as follows [ MV V M T qv M qv M qq [ V q [ CV + V CV q C qv Cqq [ V [ F τ (9) (11) with F the external wrench on the ase link, expressed in coordinates Ψ (such that it is collocated with the twist V ). To compute the matrix C(Q,v) for our system, we can use the oservations that M(q) is independent of g, that S(Q,φ) is independent of q, and that S(Q,) I. Furthermore, the partial derivative of M with respect to φ is zero since M is independent of g, and the second term of (5) is only non-zero for the C V V lock of C(Q,v). 33 The precise computational details of the partial derivatives follow the same steps as in the classical approach [1. To compute the partial derivatives of the adjoint matrices, one can use a relatively simple relation. If we express the velocity of joint k as V (k 1) (k 1)k X k k for constant X k, then the following holds: Ad gij q k Ad gi(k 1) ad Xk Ad g(k 1)j for i < k j Ad gi(k 1) ad Xk Ad g(k 1)j for j < k i otherwise To prove this, we start y writing out the spatial velocity of frame Ψ k with respect to Ψ (k 1) when i < k j: (k 1) ˆX k k ˆV (k 1)k ġ (k 1)kg 1 (k 1)k g (k 1)k g k(k 1) k q k where ˆX [ : ˆXω X v. If we compare the first and the last terms, we get R (k 1)k q ˆX ω R (k 1)k, k (1) p (k 1)k q ˆX ω p (k 1)k + X v. k (13) We can use this relation in the expression for the partial derivative of Ad g(k 1)k : [ R(k 1)k ˆp Ad (k 1)k R g(k 1)k q k q k R (k 1)k +ˆp (k 1)k (k 1)k q k R q (k 1)k q [ [ k ˆXω ˆXv R(k 1)k ˆp (k 1)k R (k 1)k ˆXω R (k 1)k ad Xk Ad g(k 1)k (1) It is now straight forward to show that Ad gij q k Similarly when j < k i. Ad g(k 1)k Ad gi(k 1) Ad gkj q k Ad gi(k 1) ad Xk Ad g(k 1)k Ad gkj Ad gi(k 1) ad Xk Ad g(k 1)j. (15) C. Manipulator dynamics on a forced non-inertial ase We now simplify and specialize the dynamics equations y assuming that the motion of the platform is fully determined y external forces that are neither known nor of interest. We only assume that the relative pose g, velocity V, and acceleration V of the ase relative to the inertial world are known from measurements. This implicitly implies that the torques applied to the internal root joints do not influence the motion of the platform, which is a reasonale assumption in our application of a relatively small root attached to a large moving ase. Since we are not interested in the external forces on the mechanism, we can consider just the second lock-row of (11), which expresses the root accelerations q as a function

of the joint torques τ as well as the non-inertial motion of the ase. This can e rewritten as M qq q + C qq + M qv V + C qv V τ (1) inertial forces which partially separates the usual root dynamics (first two terms) from the inertial forces (third and fourth term), although the matrix C qq generally still depends on V. For a static ase frame (V ), the equations reduce to the regular dynamics of an n-link rootic mechanism. Note that for constant V, the terms due to the non-inertial ase motion generally do not drop out, since a constant twist can also contain (non-inertial) angular components. Note also that neither the inertia of the ase nor the second term in (5) appear in these equations. The terms C qv and C qq can e written more explicitly as C qv C qq k1 k1 M qv q k k 1 T q M qq q k k 1 T q ( [MV V M T qv ( [MqV M T qq [ V [ V ) ) This approach can e used to otain the dynamics equations for an aritrary n-link mechanism attached to a non-inertial ase. Specific examples are presented in Section IV. D. Gravitational forces Finally we include the gravitational forces. Let the wrench associated with the gravitational force of link i with respect to coordinate frame Ψ i e given y [ [ Fg i fg Ri e ˆr gf i m i g z g ˆr gr i (17) i e z where e z [ 1 T and r i g is the center of mass of link i expressed in frame Ψ i. In our case Ψ i is chosen so that r i g is in the origin of Ψ i so we have r i g. The equivalent joint torque associated with link i is given y τ i g J i (q)ad T g i (Q)F i g(q) (1) where J i is the geometric Jacoian and Ad gi Ad g Ad gi is the transformation from the inertial frame to frame i. We note that oth R i and Ad gi depend on the ase configuration with respect to the inertial frame. The total effect of the gravity from all the links is then given y τ g n i1 τi g which enters Equation (1) the same way as the control torque. III. COMPENSATION USING MOTION PLANNING In general there are two ways to deal with the inertial forces. We can try to compensate for the effects in the controller or in the motion planning algorithm. In the first method we cancel the effects of the inertial forces in the feed-forward terms of the controller. Consider the control law τ τ ff + τ PD (19) where τ ff M qq q d + C qq d tracking terms n1 (J i Ad T g i F i g) } {{ } gravity compensation τ PD K P (q d q) + K D ( d ) PD-controller + M qv V + C qv V compensation for inertial forces () (1) This is the standard augmented PD control law which in our case also compensates for the inertial forces. As we are mainly interested in the feed-forward terms, we will assume perfect tracking, i.e. q(t) q d (t). With this control law the non-inertial and gravitational terms are regarded as disturances and are canceled. When large inertial forces are present, canceling these terms may e very energy demanding. Thus, instead of regarding these terms as disturances, we will find the trajectory for which the non-inertial and gravitational terms coincide with the tracking terms to an as large extent as possile. In doing this, the inertial forces will contriute to the desired motion instead of working against it. This will reduce the wear and tear on the manipulator, require less actuator torques and allow more accurate manipulation. Given the dynamic equations, the initial position q, and desired end position q des in joint coordinates, we want to find the optimal trajectory given y the minimum of the cost function P, i.e. T1 P min min P(τ)dt () q(t) tt where P(τ) is some cost function representing the torque required for the motion, q(t ) q, q(t 1 ) q des, (3) are the vectors descriing the initial and end positions of all the joints and M qq q + C qq + M qv V + C qv V τ g τ () determines the dynamics of the system. The gloal solution to this prolem is generally very complex. Assuming g, V (t) and V (t) known, we first need to compute M qq (q), Cqq (q,,v ), M qv (q) and C qv (q,,v ). Then we need to find the optimal trajectory (q(t), (t), q(t)) which requires the least amount of torque. Both these operations are computationally very demanding. IV. EXAMPLES We now present specific examples of how the previous modeling and planning methods can e applied in case of specific root motion ojectives and given ase motion. Here, we make specific choices as to how to discretize and approximate the prolem to make it solvale; future work will investigate different and more general approaches. 333

A. Parameterization of joint motion To reduce the search space, we assume that the shape of each joint trajectory is given so that we only need to find the starting time and the length of the motion for each joint. We also consider a cost function P(τ) that is quadratic in τ and thus reduce the prolem to T1 P min min τ T Dτ dt (5) t,t 1 tt where D is a positive definite matrix that defines a metric in τ-space, t [ t 1, t i, t n, T are the starting times and t 1 [ t 1,1 t i,1 t n,1 T are the end times for the n joints, which can all e chosen independently, with the restriction that T t i, < t i,1 T 1 for all i and for a fixed prescried time interval (T,T 1 ). We choose sinusoidal joint motions given y q i (t) a i sin( i (t t i, )), i (t) a i i a i i cos ( i (t t i, )), q i (t) q i, + a i (t t i, ) a i i sin ( i (t t i, )), () i for t (t i,,t i,1 ) and q i (t) constant otherwise. The oundary conditions q i (t i, ) q i, and q i (t i,1 ) q i,des give rise to the following two equations a i (q i,des q i, ) i π i π t i,1 t i, (7) C. 1-DoF manipulator Consider first a mechanism with a single 1-DoF prismatic joint located at p 1 [ l 1 T in Ψ and moving in the direction of the y-axis. Let the ase motion e given as in (). We set m 1 and the dynamics equations reduce to τ dt 15 1 5 τ q l 1 φ q φ g sin(φ). (9) 1.5 1.5.5 1 1.5 t Fig.. The torques needed to move the prismatic joint from q to q des 1.5 for high frequency ase motion φ(t) 1/5 sin (t) plotted with respect to the start time t and the motion length t t 1 t. The minima found are marked with an X... 1. 1. 1. 1. 1. t and hence the motion is fully parameterized y t i, and t i,1 for given q and q des. The motion planning prolem is thus reduced to finding the optimal time intervals (t i,,t i,1 ) for all joints i 1,...,n. B. Base motion The environmental disturances that affect the platform motion are wind, waves and ocean currents. The ocean currents are low frequency disturances and will not affect the manipulator dynamics. It is common to assume the principle of superposition when considering wave and wind disturances [1 and they are normally modeled in the frequency spectrum. Many good models of the ship motion for different sea states are availale in literature [1, [17. The platform motion is modeled as g (t) SE(3). Large marine vessels are often found to have a characteristic motion which we can represent as a vector suspace of se(3). For the purpose of this paper, we will estimate the main angular motion of the platform somewhat roughly y a sinusoidal motion in SO(). Assume that the waves hit the platform with a velocity in the direction of the y-axis in Ψ. The platform pose and acceleration are then given y an angular motion aout the x-axis: φ Asin (Bt) () This is a simplified motion, ut the dynamic equations are valid for any motion in SE(3). Specific examples of how this ase motion affects the manipulator motion are shown in the following. 33 Fig. 3. Optimal and worst case trajectories. The required torque is the total torque required for the desired joint motion, which is the sum of the actuator and the inertial torques. The actuator torque is the torque applied y the actuator so that the total torque is equal to the required torque, i.e. the inertial torques sutracted from the required torque. This is the torque to e minimized. The optimal interval is found at t [-1.5 -.9. The worst case is found at t [-.95 -.35. The integrated torque (squared) for the optimal solution is.7 and 11.3 for the worst case. As the length of the motion increases ( t increases in Figure ), the integrated torque increases unounded, thus the worst case starting point search is performed with a fixed motion length of.s. We start y approximating the ase motion given in () y the Taylor approximation sin (x) x x3 3! + x5 5! x7 7! + O(x9 ). (3)

We can write φ(t), φ(t) and φ(t) as ) φ(t) A (Bt (Bt)3 + (Bt)5 (Bt)7 + O(t 9 ), 3! 5! 7! ) φ(t) AB (1 (Bt) + (Bt) (Bt) + O(t ),!!! ( φ(t) AB (Bt) ) + (Bt)3 (Bt)5 + O(t 7 ). 1! 3! 5! This is typically a good approximation for one period of sinusoidal motion. We approximate the desired joint motion given y () in the same way. As we have only one joint we set t T and t 1 T 1. The minimization prolem is then reduced to t1 P min min ( q l 1 φ q φ g sin(φ)) dt (31) t,t 1 t which after sustituting the Taylor approximations reduces to the prolem of finding the minimum of a polynomial equation. We can now quickly find the optimal solution with respect to the start and end times t and t 1. We define the search space as the time interval for which the Taylor approximation is accurate, i.e., (t,t 1 ) ( t T,t T ) where t T is the wave period of the principal frequency of the waves. Figure illustrates the value of the integral (31) for different start and end times for φ(t) 1/5sin (t), q and q des 1.5. The optimal and worst case trajectories are shown in Figure 3. D. -DoF manipulator The previous example can e solved efficiently as it reduces to finding the minimum of a polynomial equation. As a second example, we show how numerical methods can e used to compute optimal motion paths for the - DoF manipulator shown in Figure 1 with realistic mass and inertia parameters. We now use the exact equations for ase motion and the manipulator dynamics. The ase moves along the angular motion pattern () at a relatively low frequency, which means the inertial forces mostly enter through a changing direction of gravity. We solve the motion planning prolem y numerically minimizing the ojective function () and parameterize the prolem as follows: each joint trajectory is given y a separate sinusoidal motion () with parameters t i, and t i,1, the total motion from start to goal is to e finished within a fixed prescried time interval (T,T 1 ) (,1), and the cost function is chosen as (5) with D 1 I and integrated over the fixed time interval (T,T 1 ). We choose the start and end configurations as q() [.5 T q(t) [.5 π π π T and the ase motion as () with A 5 π and B π 5. The motion planning prolem thus amounts to finding the eight parameters (one start and end time for each joint) that minimize the total squared required torque integrated over a t t t t t t 1 Fig.. Still shots from the simulation of the fully optimized trajectory corresponding to the solid lines in Figure 5. fixed time interval while starting and finishing the root in the required configurations. Figures and 5 show the solution otained using Matla s constrained minimization function fmincon. A full animation of the resulting motions can e found in the video accompaniment to this paper. Figure shows still shots of the optimal solution and illustrates the sinusoidal motion of the ase. Figure 5 compares three solutions: one aseline solution that simply takes the start time for each joint trajectory at t i, T and the finish time at t i,1 T 1, one solution that optimizes the cost function assuming zero ase motion (a horizontal stationary ase), and one solution that optimizes the cost function taking the real ase motion into account. The associated costs are 19, 1, and 11, respectively. The figure shows that, for this example, taking the ase motion into account can significantly reduce the cost and hence the required torque. The joint motions optimized for a static ase (dashed line) even perform worse than the aseline joint motions (dotted line) when applied during non-static ase motion. When optimizing the joint motions while taking the ase motion into account (solid line), the result is much improved, and the enefit of the resulting motions can e understood intuitively: the prismatic motion is delayed and shortened as to optimally use the changing gravity direction (due to ase rotation) in the acceleration and deceleration 335

q 1 1.5 q 1.5 1.5 q 3 1.5 1.5 q 1.5 (a) Joint trajectories (m,rad). 1 1 1 1 τ 1 τ 5 τ 3 5 1 τ () Required joint torques (N,Nm). Fig. 5. Optimal motion trajectory for a -DoF manipulator. Three different trajectories are shown: a aseline trajectory with maximum motion duration (dotted lines), an optimized trajectory assuming zero ase motion (dashed lines), and an optimized trajectory taking the correct non-zero ase motion into account (solid lines). phase, similar to the previous 1-DoF example. The resulting required actuator torque τ 1 is thus reduced to close to zero during the motion, i.e., in the time interval (t 1,,t 1,1 ) (3.,7.). Similarly, the motion of joints 3 and is delayed as to minimize the amount of time spent holding up the links against gravity. The example shows how knowledge of the ase motion can e used to significantly reduce torque requirements, even with only little freedom in the optimization (only t i, and t i,1 can e optimized). If the shapes of the trajectories are allowed to e changed and optimized in more detail, improvement should e even more significant. Including more parameters makes the optimization prolem more complex, though, and numerical solutions may get more easily trapped in local minima. V. CONCLUSIONS The classical dynamics equations for a serial manipulator have een extended to also include the motion of a forced non-inertial ase. The dynamics equations are derived using a generalized Lagrangian method. This allows us to model the ase motion as a free motion joint serially connected with the 1-DoF joints of the manipulator. Examples for a 1-DoF and -DoF manipulator mounted on a platform are presented. We include the platform motion in the dynamics and find the trajectory that takes the manipulator from an initial position to an end position with the least amount of torque and compare this with the optimal trajectory when the platform is assumed not to e moving. The simulations show that when the ship motion is known the amount of torque needed for a given task can e sustantially reduced if the inertial forces are taken into account. A possile extension for future work is to optimize the shape of the joint trajectories with more variales. Adding more details to the joint trajectories should increase performance even more. If a sufficiently accurate model of the platform can e otained, this may allow us to compensate 1 1 1 1 for the inertial forces in high accuracy applications. Another interesting topic for future work is to look into how model predictive control can e used to compensate for the platform motion. ACKNOWLEDGMENTS The first and third authors wish to acknowledge the support of the Norwegian Research Council and the TAIL IO project for their continued funding and support for this research. 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