Non-Linear Control of Manipulators
Robot Control System
2025
Robot Control System Non-Linear Control of Manipulators 2025 1 / 20
Outline
1 Introduction
2 MULTI-INPUT, MULTI-OUTPUT CONTROL SYSTEMS
3 Control Design
Robot Control System Non-Linear Control of Manipulators 2025 2 / 20
Introduction to Nonlinear Control
Solution of nonlinear systems are difficult to obtain.
Local linearization can be used to derive approximations of nonlinear
equations in the neighborhood of an operating point
Manipulator control problem not well suited to this approach because
manipulators constantly move among regions of their workspace
widely
Move the operating point with the manipulator as it moves, always
linearizing about the desired position of the manipulator
The result of this sort of moving linearization is a linear, but
time-varying, system
A more complicated control law, in which the gains are time-varying
is used
It uses a nonlinear control term to ”cancel” a nonlinearity in the
controlled system, so that the overall closed loop system is linear
Robot Control System Non-Linear Control of Manipulators 2025 3 / 20
The servo law for a multidimensional system becomes
F ′ = Ẍd + kv Ė + kp E
where and are now nxn matrices, which are generally chosen to be
diagonal with constant gains on the diagonal. E and E are nx1
vectors of the errors in position and velocity.
Robot Control System Non-Linear Control of Manipulators 2025 4 / 20
Control Law Design for Nonlinear Spring
Controller reads sensor input and computes actuator output
Figure: Closed Loop System
Robot Control System Non-Linear Control of Manipulators 2025 5 / 20
Consider nonlinear friction characteristics shown:
Figure: Force-Velocity characteristics
nonlinear coloumbian friction is described by f = bc sgn(ẋ)
Robot Control System Non-Linear Control of Manipulators 2025 6 / 20
The open loop equation is
mẍ + bc sgn(ẋ) + kx = f
The partitioned control law is f = αf ′ + β
where α = m, β = bc sgn(ẋ) + kx and f ′ = x¨d + kv ė + kp e
Robot Control System Non-Linear Control of Manipulators 2025 7 / 20
Example : Consider the single-link manipulator shown in Fig. It has one
rotational joint. The mass is considered to be located at a point at the
distal end of the link, and so the moment of inertia is ml2. There is
Coulomb and viscous friction acting at the joint, and there is a load due to
gravity.
Figure: An inverted pendulum or a one-link manipulator
Robot Control System Non-Linear Control of Manipulators 2025 8 / 20
The model of the manipulator is,
τm = ml 2 θ̈ + v θ̇ + csgn(θ̇) + mlgcos(θ) (1)
The control system has two parts, the linearizing model-based portion and
the servo-law portion. The model-based portion of the control is
f = αf ′ + β.
α =ml 2
β =v θ̇ + csgn(θ̇) + mlgcos(θ) (2)
f ′ =θ¨d + kv ė + kp e
The values of the gains for servo portion are calculated from some desired
performance specification.
Robot Control System Non-Linear Control of Manipulators 2025 9 / 20
The method to design a nonlinear controller is
Compute a nonlinear model-based control law that ”cancels” the
nonlinearities of the system to be controlled.
Reduce the system to a linear system that can be controlled with the
simple linear servo law developed for the unit mass.
The linearizing control law implements an inverse model of the
system being controlled.
The nonlinearities in the system cancel those in the inverse model;
this, together with the servo law, results in a linear closed-loop
system.
We must know the parameters and the structure of the nonlinear
system.
Robot Control System Non-Linear Control of Manipulators 2025 10 / 20
MIMO Systems
A vector of desired joint positions, velocities, and accelerations is
given, and the control law must compute a vector of joint-actuator
signals
Partitioning the control law into a model-based portion and a servo
portion, it now appears in a matrix—vector form
The control law takes the form F = αF ′ + β
Where, for a system of n degrees of freedom, F , F ′ , and are nxl
vectors
α is an nxn matrix that is chosen to decouple the n equations of
motion
If α and β are correctly chosen, then, from the F’ input, the system
appears to be n independent unit masses
The model-based portion of the control law is called a linearizing and
decoupling control law
Robot Control System Non-Linear Control of Manipulators 2025 11 / 20
THE CONTROL PROBLEM FOR MANIPULATORS
The rigid-body dynamics has the form
Figure: Block diagram representation of a robot control system.
Typically, a trajectory can be generated by solving following differential
equation,
τ = M(θ)θ̈ + V (θ, θ̇) + G (θ)
Robot Control System Non-Linear Control of Manipulators 2025 12 / 20
Robotic Manipulator
where M(θ) is the nxn inertia matrix of the manipulator, V (θ, θ̇), is
an nx1 vector of centrifugal and Coriolis terms, and G (θ) is an nx1
vector of gravity terms
Each element of M(θ) and G (θ) is a complicated function that
depends on (θ), the position of all the joints of the manipulator
Each element of V (θ, θ̇) is a complicated function of both θ and θ̇
Incorporate a model of friction which is a function of joint positions
and velocities, above equation takes the form
τ = M(θ)θ̈ + V (θ, θ̇) + G (θ) + F (θ, θ̇)
Robot Control System Non-Linear Control of Manipulators 2025 13 / 20
Partitioned control law can be used to control the system using
controller design τ = ατ ′ + β
Where τ is the nx1 vector of joint torques
If we choose α = M(θ) and β = V (θ, θ̇) + G (θ) + F (θ, θ̇)
With servo law
τ ′ = θ̈d + kv Ė + kp E
Where E = θd − θ
The closed-loop system is characterized by the error equation
Ëd + kv Ė + kp E = 0
The matrices Kp and Kv are diagonal,
Robot Control System Non-Linear Control of Manipulators 2025 14 / 20
Figure: A model-based manipulator-control system.
Robot Control System Non-Linear Control of Manipulators 2025 15 / 20
The ideal performance represented by (14) is unattainable in practice
because,
The discrete nature of a digital-computer implementation, as opposed
to the ideal continuous-time control law
Inaccuracy in the manipulator model
Robot Control System Non-Linear Control of Manipulators 2025 16 / 20
CURRENT INDUSTRIAL-ROBOT CONTROL SYSTEMS
An industrial robot with a high-performance servo system is shown in Fig.
Figure: The Adept One, a direct-drive robot by Adept Technology, Inc.
Robot Control System Non-Linear Control of Manipulators 2025 17 / 20
Individual-joint PID control
Most industrial robots nowadays have a control scheme described by
α =I
β =0
Z (3)
′
τ =θ¨d + Kv Ė + Kp E + Ki Edt
Here Kp , KV and Ki are diagonal matrices.
This type of PID control scheme is simple because each joint is
controlled as a separate control system.
One microprocessor per joint is used to implement control law.
The performance of a manipulator controlled in this way is not simple
to describe.
Robot Control System Non-Linear Control of Manipulators 2025 18 / 20
No decoupling is being done, so the motion of each joint affects the
other joints. These interactions cause errors, which are suppressed by
the error-driven control law.
It is impossible to select fixed gains that will critically damp the
response to disturbances for all configurations. Therefore, ”average”
gains are chosen, which approximate critical damping in the center of
the robot’s workspace.
It is important to keep the gains as high as possible, so that the
inevitable disturbances will be suppressed quickly.
Effect of gravity is nullified by adding compensation term in the
control law.
Robot Control System Non-Linear Control of Manipulators 2025 19 / 20
LYAPUNOV STABILITY ANALYSIS
Robot Control System Non-Linear Control of Manipulators 2025 20 / 20
