ROBOTICA 03CFIOR 03CFIOR Basilio Bona DAUIN – Politecnico di Torino Basilio Bona 1 ROBOTICA 03CFIOR
Control – Part 1
Introduction to robot control � The motion control problem consists in the design of control algorithms for the robot actuators � In particular it consists in generating the time functions of the generalized actuating torques, such that the TCP motion generalized actuating torques, such that the TCP motion follows a specified task in the cartesian space, fulfilling the specifications on transient and steady-state response requirements Basilio Bona 3 ROBOTICA 03CFIOR
Tasks Two types of tasks can be defined: 1. Tasks that do not require an interaction with the environment ( free space motion ); the manipulator moves its TCP following cartesian trajectories, with constraint on positions, velocities and accelerations due to the manipulator itself or the task requirements q t q t ( ) ( ) � Sometimes it is sufficient to move the joints from a specified value � Sometimes it is sufficient to move the joints from a specified value i 0 q t ( ) to another specified value without following a specific geometric i f path 2. Tasks that require and interaction with the environment, i.e., where the TCP shall move in some cartesian subspace while subject to forces or torques from the environment We will consider only the first type of task Basilio Bona 4 ROBOTICA 03CFIOR
Motion control � In particular the motion control problem consists in generating the time functions of the generalized actuating torques, such that the TCP motion follows a specified task in the cartesian space, fulfilling the specifications on transient and steady-state response requirements � Control schemes can be developed for: � Joint space control � Task space control considering that the task description is usually specified in the task space, while control actions are defined in the joint space Basilio Bona 5 ROBOTICA 03CFIOR
Joint space control The Inverse Kinematics block transforms the desired task space positions The Inverse Kinematics block transforms the desired task space positions and velocities into desired joint space reference values. The Transducer measures the value of the joint quantities (angles, displacements) and compares them with the desired ones, obtained, if necessary, from the desired cartesian quantities. The Controller uses the error to generate a (low power) signal for the Actuator that transforms it in a (high power) torque (via the Gearbox ) that moves the robot joints Basilio Bona 6 ROBOTICA 03CFIOR
Task space control p p d Controller Actuator Gearbox Robot Transducer In this case, the Transducer must measure the task space quantities in order to In this case, the Transducer must measure the task space quantities in order to compare them with desired ones. Usually this is not an easy task, since it requires environment-aware sensors; the most used ones are digital camera sensors (vision-based control) or other types of exteroceptive sensors (infra-red, ultra-sonic, ...). Otherwise one uses the direct kinematics to estimate the task space pose Torques are always applied to the joints, so Inverse Kinematics is hidden inside the Controller block Basilio Bona 7 ROBOTICA 03CFIOR
Joint space control architectures Two main joint space control architectures are possible Decentralized control or independent joint control : each i -th joint motor has a local controller that takes into account only local variables, i.e., the joint ɺ q t ( ) q t ( ) position and velocity i i The control is of SISO type, usually based on a P, PD or PID architecture The controller is designed considering only an approximated model of the i- th joint. This scheme is very common in industrial robots, due to its simplicity, This scheme is very common in industrial robots, due to its simplicity, modularity and robustness The classical PUMA robot architecture is shown in the following slide Centralized control : there is only one MIMO controller that generates a command vector for each joint motor; it is based on the complete model of the manipulator and takes into account the entire vector of measured positions and velocities Basilio Bona 8 ROBOTICA 03CFIOR
Decentralized control Decentralized Joint q t 1 ( ) joint 1 Control reference controller 1 q t 2 ( ) joint 2 Task space Joint space reference controller 2 … q t 6 ( ) joint 6 reference controller 6 Basilio Bona 9 ROBOTICA 03CFIOR
Decentralized control Teach Terminal Disk Other pendant PUMA Control Amplifier µ G D/A Motor 1 DLV-11J Encoder T=0.875 ms EPROM EPROM Reference Interface angles T=28 ms RAM CPU µ G D/A Amplifier Motor 6 Encoder T=0.875 ms COMPUTER ROBOT CONTROL Basilio Bona 10 ROBOTICA 03CFIOR
Motor and gearbox model (rigid body assumptions) Gearbox = Geartrain N Friction r = N Gearbox ′ N; ω τ , Inertia m r Robot τ τ m m Friction ′ ′ ′ Inertia N ; ω τ , m r Motor ′ τ m ′ ′ τ ω τ ω Output power Input power r m r m Basilio Bona 11 ROBOTICA 03CFIOR
Motor and gearbox model ω β m b τ = τ + τ r m p Γ R L τ N i a a b m a i i e Γ ′ N v E m τ a ′ τ p m ′ ω β m m ′ ′ ′ τ = τ − τ r m p ′ τ p Basilio Bona 12 ROBOTICA 03CFIOR
Losses in geared motor Motor Joint side side ′ ′ ω τ ω τ ω τ v i Ei a a a m r m m Armature Motor m r Joint Gearbox circuit inertia inertia d d η + + L L i i R i R i a a a a d t gearbox efficiency d voltage drop ′ ′ Γ ω + β ω d m m m m d t Γ ω + β ω b m b m d t τ ′ p τ p Basilio Bona 13 ROBOTICA 03CFIOR
Gearbox model ρ ′ ′ = N ρ θ ρθ = = r ′ ′ ω τ ρ N ′ ′ = ρ ω ρω m r Joint side ρ η = 1 Ideal gearbox: θ Power in Power out GEARBOX GEARBOX θ ′ ′ ′ ω τ ′ ′ ω τ = ηω τ m r m r m r ρ ′ Motor side ′ ′ ω τ ′ τ = τ m r r r r ′ ω ω = m m r Basilio Bona 14 ROBOTICA 03CFIOR
CC motor equations – 1 MOTOR SIDE d = − − L i v R i E a a a a a d t φ = K i φ e ′ ′ = φ φ ω ω = ω ω E k K ′ ′ ω ω ≈ ≈ ⇒ ⇒ ≈ ≈ m m m m k k k k K K K K ′ ′ ω τ τ = φ = k i K i τ m a a ′ τ = i m a K τ ɺɺ ɺ ′ ′ ′ τ = Γ θ + β θ = Γ ω + β ω ɺ p m m m m m m m m ′ ′ ′ τ = τ − τ r m p Basilio Bona 15 ROBOTICA 03CFIOR
CC motor equations – 2 MOTOR SIDE JOINT SIDE 1 ′ τ = τ r ′ = τ τ r r r r r ′ ′ = τ − τ r ( ) 1 ( m p = τ + τ ) ′ ′ ′ = τ − Γ ω − β ω ɺ r m p r m m m m m 1 ′ = τ − Γ ω − β ω r r ɺ r ( ) ( ) = = τ τ + + Γ Γ ω ω + + β ω β ω ɺ ɺ m m m m m m m m m m m m b b m m b b m m r r ′ 2 ( = τ − Γ ω ɺ + β ω r r ) m m m m m 1 1 1 ′ ′ = τ + Γ ω + β ω ɺ m b m b m r r r 1 1 ( ′ ′ = τ + Γ ω + β ω ɺ ) m b m b m r 2 r Basilio Bona 16 ROBOTICA 03CFIOR
Control equations ɺɺ + ɺ ɺ + ɺ + + T = τ H q q ( ) C q q q ( , ) B q q ( ) g q ( ) J F e component-wise joint torques n n n ∑ ∑ ∑∑ ∑∑ + + + + β β + + + + τ τ = = τ τ q q ɺɺ ɺɺ q q ɺ ɺ ɺ ɺ q q H H ( ) ( ) q q h h ( ( ) ) q q q q q q g g ( ( ) ) ij ij j j ijk ijk j j k k bi bi i i i i fi fi ri ri = = = j 1 j 1 k 1 n n n ∑ ∑∑ ɺɺ + ɺɺ + ɺ ɺ + β ɺ + + τ = τ q q q q H ( ) q H ( ) q h ( ) q q q g ( ) i i i ij j ijk j k bi i i fi r i ≠ = = j i j 1 k 1 Coriolis & centripetal Friction, gravity Inertial torques torques & external torques Basilio Bona 17 ROBOTICA 03CFIOR
Control equations n n n ∑ ∑∑ ɺɺ + ɺɺ + ɺ ɺ + β ɺ + + τ = τ H ( ) q q H ( ) q q h ( ) q q q q g ( ) q ii i ij j ijk j k bi i i fi ri ≠ = = j i j 1 k 1 q ɺɺ + β ɺ + τ + τ + τ + τ = τ H ( ) q q ii i bi i M i ci gi f i r i Modelled torques “Disturbance” torques Basilio Bona 18 ROBOTICA 03CFIOR
From single motor model to robot control equation ′ ′ ′ θ ω ω ɺ = ɺ = ɺ ɺ = q mi q mi q mi i i i r r r i i i Gearbox transformation n + ∑ ∑ τ τ = = ɺɺ ɺɺ + + β β ɺ ɺ + ɺɺ ɺɺ + + τ τ + + τ τ + + τ τ + + τ τ H q H q q q H q H q ri ri ii ii i i bi bi i i ij ij j j Mi Mi ci ci gi gi fi fi ≠ j i ′ ′ ω ω ɺ = Γ + β + τ + τ + τ + τ mi mi Structured bi bi Mi ci gi fi r r disturbance i i ′ ′ ω ω ɺ = Γ + β + τ mi mi bi bi di r r i i Equation seen at the joint side Basilio Bona 19 ROBOTICA 03CFIOR
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