A unified control architecture for navigation of nonholonomic systems
ABSTRACT The paper presents a unified control architecture for motion planning and navigation of constrained systems. It provides a systematic approach for planning any motion that may be specified by equations of algebraic or differential constraints. It is based upon one dynamic control model for constrained systems, which is not sensitive to the constraint kind and order. The preplanned reference motion may be executed by nonlinear control algorithms.
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ABSTRACT: The subject of the paper is the motion control problem of wheeled mobile robots (WMRs) in environments without obstacles. With reference to the popular unicycle kinematics, it is shown that dynamic feedback linearization is an efficient design tool leading to a solution simultaneously valid for both trajectory tracking and setpoint regulation problems. The implementation of this approach on the laboratory prototype SuperMARIO, a two-wheel differentially driven mobile robot, is described in detail. To assess the quality of the proposed controller, we compare its performance with that of several existing control techniques in a number of experiments. The obtained results provide useful guidelines for WMR control designers.IEEE Transactions on Control Systems Technology 12/2002; · 2.00 Impact Factor
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ABSTRACT: In order to avoid wheel slippage or mechanical damage during the mobile robot navigation, it is necessary tosmoothly change driving velocity or direction of the mobile robot. This means that dynamic constraints of the mobile robotshould be considered in the design of path tracking algorithm. In the study, a path tracking problem is formulated asfollowing a virtual target vehicle which is assumed to move exactly along the path with specified velocity. The drivingvelocity control law is designed basing on bang-bang control considering the acceleration bounds of driving wheels. Thesteering control law is designed by combining the bang-bang control with an intermediate path called the landing curve whichguides the robot to smoothly land on the virtual target''s tangential line. The curvature and convergence analyses providesufficient stability conditions for the proposed path tracking controller. A series of path tracking simulations and experimentsconducted for a two-wheel driven mobile robot show the validity of the proposed algorithm.Journal of Intelligent and Robotic Systems 01/1999; 24:367-385. · 0.83 Impact Factor
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ABSTRACT: This thesis explores the paradigm of two degree of freedom design for nonlinear control systems. In two degree of freedom design one generates an explicit trajectory for state and input around which the system is linearized. Linear techniques are then used to stabilize the system around the nominal trajectory and to deal with uncertainty. This approach allows the use of the wealth of tools in linear control theory to stabilize a system in the face of uncertainty, while exploiting the nonlinearities to increase performance. Indeed, this thesis shows through simulations and experiments that the generation of a nominal trajectory allows more aggressive tracking in mechanical systems. The generation of trajectories for general systems involves the solution of two point boundary value problems which are hard to solve numerically. For the special class of differentially flat systems there exists a unique correspondence between trajectories in the output space and the full state and input space. This allows us to generate trajectories in the lower dimensional output space where we don't have differential constraints, and subsequently map these to the full state and input space through an algebraic procedure. No differential equations have to be solved in this process. This thesis gives a definition of differential flatness in terms of differential geometry, and proves some properties of flat systems. In particular, it is shown that differential flatness is equivalent to dynamic feedback linearizability in an open and dense set. This dissertation focuses on differentially flat systems. We describe some interesting trajectory generation problems for these systems, and present software to solve them. We also present algorithms and software for real time trajectory generation, that allow a computational tradeoff between stability and performance. We prove convergence for a rather general class of desired trajectories. If a system is not differentially flat we can approximate it with a differentially flat system, and extend the techniques for flat systems. The various extensions for approximately flat systems are validated in simulation and experiments on a thrust vectored aircraft. A system may exhibit a two layer structure where the outer layer is a flat system, and the inner system is not. We call this structure outer flatness. We investigate trajectory generation for these systems and present theorems on the type of tracking we can achieve. We validate the outer flatness approach on a model helicopter in simulations and experiment.
2011 American Control Conference
on O'Farrell Street, San Francisco, CA, USA
June 29 - July 01, 2011
Abstract— The paper presents a unified control architecture
for motion planning and navigation of constrained systems. It
provides a systematic approach for planning any motion that
may be specified by equations of algebraic or differential
constraints. It is based upon one dynamic control model for
constrained systems, which is not sensitive to the constraint
kind and order. The preplanned reference motion may be
executed by nonlinear control algorithms.
onstrained nonholonomic systems require nonlinear
control methods since their linearized control models
are usually not controllable . A nonlinear control
design process consists of three basic steps: model building,
a controller design and its implementation. Usually modeling
and control design are related to constraints on a system.
In the paper we consider control oriented dynamic
modeling and a controller design for constrained systems.
We show that in the modeling step, we may obtain a unified
dynamic model suitable for designing controllers despite of
the kind and order of constraints imposed on a system.
The first motivation for this research is that mechanical
systems are subjected to material and non-material
constraints. The latter ones are task, control and design
based, and they may be specified by differential equations of
high order. We refer to them as programmed constraints
[2,3]. They are non-material since they may be put by a
designer like a trajectory to follow, which is specified by an
algebraic equation but is not treated as a constraint in control
setting [1,4]. The trajectory is either given a priori or by a
motion planner and next it is passed to a controller [5-7]. An
industrial manipulator, holonomic by its nature, may become
nonholonomic when constraints are imposed upon its motion
properties . A space vehicle is nonholonomic due to the
conservation of its angular momentum. Also, a leader-
follower system that consists of a couple of robots is a
noholonomic system dedicated to navigate towards task
based missions, which are not treated as constraints on
motion [10-13]. Then, there was no unified constraint
formulation for control applications. An exception is the
second order nonholonomic constraint due to an unactuated
degree of freedom .
Secondly, a control framework that incorporates a system
dynamics, i.e. model-based, is developed on traditional two-
Manuscript received September 20, 2010.
E. M. Jarzębowska is with the Warsaw University of Technology,
Institute of Aeronautics and Applied Mechanics, 00-665 Warsaw,
Nowowiejska 24 str., POLAND (phone: +48 222346029; fax: +48
226282587; e-mail: email@example.com).
level tracking control architecture for nonholonomic
systems. The lower control level operates within a kinematic
model to stabilize a system motion to a desired trajectory.
The upper control level uses a dynamic model and stabilizes
feedback obtained on the lower control level [12,14]. The
underlying dynamics is based on the Lagrange approach, so
first order nonholonomic constraints may be merged into it.
Finally, latest results in modeling constrained systems
showed that material and programmed constraints might be
presented in a unified constraint formulation suitable to
control design [9,15]. This is in contrast to classical
analytical mechanics that offers methods of the generation of
dynamic models of systems with first and second order
nonholonomic constraints . Constraints on motion
specified by equations of high order could not be included
into these dynamic models. The Lagrange approach is used
the most often for model building in control. It is not suitable
then due to the constraint order it may incorporate and the
reduction procedure that has to be performed. Thus, there
were neither systematic nor unified approaches to modeling
systems with constraints of order higher than one.
A unification of modeling constrained systems, in both
kinematic and dynamics settings are presented in the paper.
It yields the generalized programmed motion equations
(GPME) that may capture systems with high order
constraints . It results in a unified control oriented models
of constrained systems and design of a new control strategy.
II. A UNIFIED SPECIFICATION OF CONSTRAINED SYSTEMS
A. Sources of Constraints on Control Systems
In mechanics, a type of a nonholonomic constraint arises
from a condition of rolling without slipping. It is first order
and of the material type. For space vehicles, a first order
nonholonomic constraint results from the conservation of the
angular momentum but it is referred to as the conservation
law not as a kinematic constraint [16,17]. In control, there
are more constraint sources. A wheeled vehicle undergoes
motion constraints that depend on its design, its interaction
with the environment, control design and task specifications
[1,4]. They are not treated as constraints. However, they may
be regarded as non-material constraints, i.e. in control setting
types of constraints may be as follows [3,15]:
1. Material constraints .
2. Conservation laws [16,17].
3. Design constraints – they may arise from bounded linear
and angular velocities, the lateral acceleration, or from a
bounded trajectory curvature for wheeled vehicles [1,4].
A Unified Control Architecture for Navigation of Nonholonomic
Elżbieta M. Jarzębowska, Member IEEE
978-1-4577-0079-8/11/$26.00 ©2011 AACC 1714
4. Control constraints – they arise mostly from the limitation
of a number of control inputs [8,15].
5. Programmed constraints – they may arise from task and
requirement specifications put by designers [1,4,9,15].
B. Control Oriented Constraint Formulation
The idea is to develop a unified constraint formulation,
which may include the constraint types listed above, and a
unified dynamic model of a system with such constraints.
The constraint formulation is proposed to be [2,15]
where p is the constraint order, q - n-vector of generalized
coordinates, B - full rank (k n) matrix, nk and s - k-vector.
We assume that (1) are linear in p–th order derivative of
coordinates or we can transform them to this form. They may
specify both material and non-material constraints since the
type of a constraint equation does not influence the
generation of equations of motion of a system subjected to it.
For p=0 we get a configuration constraint, which may be
material and specify a constant distance between link ends or
be a programmed constraint on a trajectory. When p=1 a
constraint equation may be material and specify a condition
of rolling without slipping. However, it may arise from the
conservation law or be a programmed constraint on a desired
velocity. Material constraints are of orders p=0 or p=1, the
equation of the conservation law is of order p=1, and
constraint equations for p>1 are of the non-material type.
Definition 1: The equations of the constraints (1) are
completely nonholonomic if they cannot be integrated, i.e.
cannot be presented as equations of a lower order in
If we can integrate (1) (p-1) or less times, they are
partially integrable. If (1) can be integrated completely, they
are holonomic. We assume that (1) are completely
nonholonomic. Definition 1 extends the definition of
completely nonholonomic first order and second order
constraints [8,17]. Necessary and sufficient integrability
conditions for differential equations of arbitrary order such
as (1) are formulated in .
The unified constraints (1) can be presented in the
standard state-space control form .
)(p (p)) (p
III. A REFERENCE MODEL OF A CONSTRAINED SYSTEM
A. A Dynamic Reference Model
A unified dynamic model of a system with the constraints
(1) is derived using the GPME applying the algorithm .
Assume that (1) may be solved, at least locally, with respect
to a vector
q of dependent coordinates, i.e.
, , (t
a designer, e.g. with respect to control inputs.
1. Construct a function
. The selection is due to
P such that
and T is the kinetic energy of an unconstrained system,
is its p-th order time derivative, and
2. Construct a function
R such that
R where equations (2) replace
4. Assuming that components of external forces satisfy
, the generalized programmed motion
equations of a system with the constraints (1) are
Equations (6) and (1) admit the following properties.
Property 1: Equations (6) are (n-k) second order differential
equations and together with (1) can be presented as [2,9]
) qV(q,q M(q)
where M(q) is a (n-k)n inertia matrix,
velocity dependent vector, D(q) is a (n-k)-vector of gravity
is a (n-k)-vector of external forces.
Equations (7) are a unified constrained dynamic model.
2: Equations (7) are in the reduced-state form; constraint
reaction forces are eliminated in the derivation.
Property 3: Dynamic models of systems with constraints of
order p=1, i.e. Lagrange’s based, transformed to the
reduced-state form are peculiar cases of (7) [14,16,17].
is a (n-k)-
B. A Kinematic Reference Model
When the number of both material and programmed, or
only programmed constraints is k<n, the program is partly
specified. When n=k, i.e. B is a full rank (n n) matrix the
program is fully specified. Then, instead of the unified
dynamics (7), the constraints (1) become a unified kinematic
reference model, i.e.
When the unified kinematics (8) fully specifies motion, it has
to be verified by analyzing its solutions if the constraints are
eligible for a system, i.e. if it is capable of reaching desired
positions, velocities and accelerations to follow programmed
constraints, and if they do not violate any material constraint.
IV. A UNIFIED CONTROL STRATEGY FOR TRACKING
A. Constrained Motion Planning
For control purposes we introduce definitions.
Definition 2 : The unified dynamic model (7) is a
reference dynamic model for a constrained motion, shortly
the reference dynamics.
Specialized terms to the
It is the extension of models reported in  which apply
only to holonomic and first order nonholonomic systems.
Definition 3: The unified kinematic model (8) is a reference
kinematic model for a constrained motion, shortly the
It is the extension of kinematic models applied to control,
e.g. reported in [1,17].
The reference model, either dynamic or kinematic, may be
employed to plan motion according to the constraints on a
system. The selection of the scheme of the generation of the
reference motion depends upon the constraints on a system.
Definition 4: Constrained motion planning for a system
subjected to the constraints (1) consists in finding time
histories of programmed positions
derivatives in motion consistent with the constraints.
Specifically, trajectory planning consists in obtaining a
of (7) or (8), in which a programmed
constraint equation is algebraic.
and their time
B. Constrained Motion Navigation
Originally, the reference dynamics (7) is employed to
design the model reference tracking control strategy for
programmed motion, shortly - the strategy for programmed
motion tracking [19,20]. It may be extended to encompass
the reference kinematics (8).
Fig.1. Architecture of the model reference tracking control strategy for
The control goal is as follows: Given a programmed
motion specified by the constraints (1) and the system
reference dynamics (7) or kinematics (8), design a feedback
controller to track the desired programmed motion.
Architecture of the strategy, presented in Fig. 1, is based
upon two models: the reference dynamics (7) or kinematics
(8), whose outputs are inputs to a tracking controller, and the
unified dynamic control model
τ D(q) q ) q
Equations (9) are the GPME for p=1. They consist of (n-k)
equations of motion and k equations of material constrains
and conservation laws. The matrix M(q) is then (n-k)n and
B1(q) is a full rank (kn) matrix. Since the constraints are
linear first order,
is replaced by
quantifies effects of Coriolis and centripetal forces. Other
forces can be added to the left-hand side of (9).
The following properties of (9) can be derived from
Property 4: The unified dynamic control model (9) is
equivalent to the reduced-state Lagrange equations .
Property 5: The unified dynamic control model (9) can be
presented in a standard control form by reusing the constraint
equations presented as
independent and dependent coordinates, respectively.
Columns of the matrix G(q) span the right null space of
B1(q). It is a (n m) matrix, m=n-k, and has the form
where I is a (m m) identity matrix,
smooth (km) matrix function. The matrix
matrix function, and
is a (kk) locally nonsingular
matrix function. Elimination of second order derivatives of
dependent coordinates from the first of equations (9) yields
Equations (10) are exactly the reduced-state Lagrange
equations of a nonholonomic system.
is a symmetric, positive definite (n-
k)(n-k) matrix [14,17].
Property 7: There exists a static state feedback
such that the dynamics (10) can be
transformed to the state-space control form. Indeed,
introduce a new state variable
, for which (10) takes the form
which is a desirable state-space control form. The controlled
Property 8: Based on properties 4-7, all theoretical control
results obtained for the Lagrange based control dynamics can
be applied to the unified control dynamics (7).
, where partition of q is
are vectors of
is a locally
1 B (q) is
is a k(n-k)
, (11) yields
2 x is usually a vector of controlled velocities.
The strategy is not sensitive to the constraint order. This is
in contrast to current control design approaches, in which
each constraint type requires a control strategy modification.
The strategy may be specialized in two ways. Firstly, it is
applicable to systems with completely known or uncertain
dynamics . Secondly, different control laws may be
employed to it, i.e. we may switch between controllers to
ensure a desired tracking control precision. The block of
“specialized terms to the control law” reflects these
specializations. The modular strategy architecture enables
replacing the reference dynamics (7) by the reference
kinematics (8). The strategy is developed for tracking but it
may be applied to more general tasks, e.g. to navigation
robot formations .
Main advantages of the strategy are as follows:
- The reference dynamics (7) captures high order
nonholonomic constraints on systems and enables planning
any programmed motion.
- It extends trajectory tracking to programmed motion
- The separation of programmed constraints from others
results in the unified dynamic control model (9) equivalent
to models actually used in control theory.
- The equivalence of (9) and the Lagrange based models
promotes adaptation of existing control algorithms even
these dedicated to holonomic systems.
- It uses one dynamic control model (9) to a system
subjected to the constraints (1).
- A library of reference models for different tasks can be
generated off-line and stored in a computer.
V. EXAMPLES – CONSTRAINED MOTION NAVIGATION
A. Material and Task Based Constraints
Consider a two-wheeled robot whose kinematics is
equivalent to that of a unicycle. Let be the heading angle
of the wheel, measured from the axis x and - rotation angle
due to rolling. Coordinates of the wheel contact point with
the ground are (x,y). Nonholonomic material constraints due
to rolling the wheels without slipping on a plane surface are
, 0 cos
To show the GPME based Algorithm application, consider
robot navigation along a trajectory of a specified change of
its curvature profile. It results in the constraint
2 / 3
where F0 does not contain terms with third order time
derivatives of variables. For simulations take the curvature
profile =2sint +1. Both constraints (13) and (14) are
transformed to the form (1). The reference dynamics (7) is
derived using the Algorithm for p=3. The control dynamics
(9) is derived for p=1 and it takes the material constraints
(13) into account. Assuming that only control forces act
upon the robot, its motion according to (13) and (14) is
presented in Fig. 2. The controller is the computed torque.
For the program specified by the third order differential
equation (14), the Lagrange-based approach fails [2,9].
Different task based constraints and control laws can be
applied to navigate the robot with no changes in the strategy.
Fig. 2. Programmed motion tracking for the reference motion (14) with
σ=3: reference motion (), controlled motion ().
B. Constraints on a Holonomic System
Consider a two-link planar manipulator model whose two
degrees of freedom are described by joint angles
Select the constraint (14) for the end-effector motion. In the
joint space it has the form
F do not contain third time derivatives of the
angles and include data about the end-effector trajectory
curvature , which is =0.6+0.02t. The constraint (15) may
mimic tasks like writing, scribing or painting. The reference
dynamics is generated applying the Algorithm for p=3 and
the control dynamics is developed as for any holonomic
system. Fig. 3 shows the reference motion on the (x,y) plane.
It was selected to show that the programmed motion may be
reachable for the end-effector for some time only. After
reaching the position marked by the arrow, links of a given
length cannot follow the program any more. This
demonstrates that a program formulated for a system should
be inspected via the reference motion outputs.
1 F and
-1.5-1 -0.50 0.511.52
Fig. 3. Reference programmed motion for the end-effector.
C. Constraints from an Underactuation
Consider again the manipulator model from Example B. It
is now equipped with one actuator in the first joint. A control
objective is to move the end-effector according to a
programmed motion specified by (15). The reference
dynamics for the underactuated manipulator is the same as in
Example B but its control model is
The equation for the unactuated joint is second order
is present in the inertia matrix .
From the first of equations (16)
and inserted to the first one yields
may be obtained as
) 2 (
Using the partial feedback linearizing controller
equations (16) become
Equations (18) can be expressed in the state space control
x , x
, we obtain
basis vector in
is PD with gains ks=20, kd=10. Tracking results are
presented in Fig. 4 and 5.
, 0 ,
R , are the drift and control vector fields on
) 2 / , 2/() 2/R
. The selected controller
, 0 , 0 (
, 2 /
D. Conservation Laws
Consider a model of a space manipulator. It is the same as
in Example B with a base added to it, which is described by a
moment of inertia J and - orientation angle relative to a
fixed axis. Let 1 be the angle of the first link of mass
second link of mass
m and length
Masses are concentrated at link ends. The base is pinned to
the ground at its center and it permits the body to rotate
freely but prevents translation. Holonomic constraints arising
from the linear momentum conservation in a real space
manipulator are replaced with holonomic pinned constraints.
1l relative to the base, and 2 - the angle of the
2 l relative to the first.
Fig. 4. Programmed motion tracking by the PD controller.
Fig. 5. End-effector position tracking errors ex , ey.
When the angular momentum is conserved, e.g. it is zero, it
become a nonholonomic constraint of the form
The structure of (20) is the same as the material constraint
(13). Motion planning for the space manipulator as well as
its navigation in space can be done in the same way as in
Example A; for a case of trajectory tracking see .
. 0)22 (
E. A Multibody Nonholonomic System
A leader-follower system is usually treated as a separate
control system comparing to a single robot. Let us show that
it may be modeled and controlled in the same way as other
constrained systems. Take a leader, which is the robot as in
Example A. Two followers are robots of the same
kinematics. Nonholonomic material constraints for the leader
and followers are specified by (13) and the task based by
Using the Algorithm for p=3 and the
strategy for programmed motion tracking, we obtain robot
formation navigation presented in Fig. 6.
F. A Fully Specified Program
Consider a task of navigating a unicycle from Example A
along a desired trajectory to a rest position. To this end,
supplement the constraints (13) by one equation that
specifies the trajectory, e.g.
and the second for the termination of motion after a specified
time, say 20s. Select an initial velocity
) t ( f v
. The reference kinematics is
)5 . 0t