Abstract
MARS greenhouse needs mobile robots with on-board arms, that are capable of navigating autonomously in the greenhouse, performing tasks such as carrying trays containing plants, farming, harvesting, house keeping, inspection, cleaning, health monitoring and so on. An adaptive network framework is introduced for the motion control of a mobilebase with an on-board arm using Lyapunov’s approach and it is rigorously justified for MARS Greenhouse operation scenario wherein the tray weight containing plants change considerably. Initially, a linear in the parameter (LIP) based adaptive controller is designed to estimate the unknown parameters of the mobile base plus robot arm system after the incorporation of non-holonomic constraints. Later, the proposed adaptive network approach, which relaxes several key assumptions, is shown to be applicable with any function approximator. This adaptive network approach provides an inner loop that accounts for possible motion of the on-board arm, with changing endeffector loads, while the base is carrying out a task. The case of maintaining a desired course and speed as the on-board arm moves to its desired orientation with an unknown endeffector load is considered.
1. Introduction
An autonomous system for MARS greenhouse must be capable of navigating in the amidst of fleet of autonomous robots, perform tasks such as carrying plant trays, farming, harvesting, inspection, health monitoring and so on. Demonstration of such an autonomous system is contingent upon developing coordinated control methodologies in the presence of uncertain dynamics of the base plus arm, unknown endeffector loads, and the environment. Numerous papers have been reported in recent years on the control of mechanical systems with non-holonomic constraints [1-6]. Several papers [1-6] examine the control theoretic issues, which pertain to both holonomic and non-holonomic constraints, coordinated obstacle avoidance [1] in a very general manner. Studies show that despite the controllability of the mobile robot system, pure static state-feedback stabilization of the cart around a given terminal configuration, which includes both position and orientation, is impossible [4].
Coordinated motion control problem is considered in this paper for the mobile base with an onboard arm in the presence of uncertain dynamics of the base plus arm, nonholonomic constaints, unknown loads resulting from the task, and operating in an unknown environment. Basic objectives for the vehicle are following a planned trajectory, with a desired speed, to attain a desired docking angle and at the same time move the on-board arm to a desired orientation while the vehicle is in motion. These maneuvers are necessary to pick a tray of vegetables, pluck a vegetable, and to perform other tasks such as harvesting, farming, seeding etc. In this paper, the dynamics of a mobile robot plus an on-board arm in [5], including non-holonomic constraints are considered.
An adaptive network framework is introduced for the motion control of the combined system using Lyapunov’s approach and rigorously justified. Initially, a linear in the parameter (LIP) based adaptive controller is designed to estimate the unknown parameters of the composite mobile base plus robot arm system after the incorporation of non-holonomic constraints. Later, the proposed adaptive network approach, which relaxes several key assumptions including the LIP, is shown to be applicable with any function approximator. This adaptive approach provides an inner loop that accounts for possible motion of the on-board arm. The case of maintaining a desired course and speed as the on-board arm moves to its desired orientation are considered while estimating the unknown masses of the links and the base, endeffector loads, and with the friction coefficient of the base.
The mobile base will transport plant trays (weights change significantly over time) in the MARS greenhouse from one location to the another and the trajectory errors should be small even if the arm is required to move during the base motion. The complete base arm controller has an adaptive feedback linearization inner loop, which takes into account the non-holonomic constraints and compensates for a possibly moving on-board arm, and an outer loop for either tracking or path following, thus achieving coordinated vehicle/arm motion. Section 2 presents the dynamic equations whereas Section 3 details the controller design. Section 4 proposes the neural network (NN) controller design and in Section 5, outer trajectory tracking loop is designed. Section illustrates the simulation results.
2. Dynamic Equations
The generalized coordinates for the composite base/robot arm system, shown in Fig. 2.1, are denoted by , where (x ,y) describe the position and be the heading angle of the base. For definiteness the arm is considered to have three links; let be the generalized coordinates (e.g. joint angles) of the arm. The dynamics of the mobile base with an onboard arm can be written as [5]
. (2.1)
where is given in space coordinates as the dynamics satisfy the robot arm properties. For trajectory tracking an additional inner loop is designed to convert Cartesian trajectory into space coordinates [5]. The next step is to appropriately design a control structure with a suitable set of generalized coordinates so that the interactions between the base and the on-board arm is compensated.
The dynamic equations expressed in (2.1) can be written as
, (2.2)
with
, (2.3)
. (2.4)
The objective is to determine the control torque inputs that guarantee suitable performance in terms of the motion of the mobile base; a desired course and speed . In the case where it is desired to follow a prescribed course and speed, one may define an auxiliary input v(t) using input/output feedback linearization [5] under the assumption that the dynamics are accurately known according to
. (2.5)
This cancels the non-linearities to obtain the simple input-output relation of a Newtonian system. To complete the design of the control law, it remains evidently to select v to stabilize the trajectory following error system. After performing the feedback linearization, the dynamical equations of the mobile base can be expressed as
, (2.6)
where , B = and .
To complete the design of the base steering control law, it remains evidently to select u to stabilize the trajectory following error system. Initially we deploy LIP assumption but later, this assumption is relaxed when a neural network is used.
Fig. 2.1: A Mobile Robot with an On-board Arm.
3. Tracking Problem of the Base Plus Arm
The primary goal of this paper is to track a desired output while keeping the states and control bounded. An adaptive feedback linearization approach will be used to achieve acceptable output tracking error (bounded-error tracking) while all the states and controls remain bounded. Thus, an adaptive controller will be designed that effectively feedback linearizes (2.1). To this end we will make some mild assumptions that are widely used and hold in any practical design. First define a desired vector that satisfies the following assumption:
Assumption 1 : The desired trajectory vector xd is assumed to be continuous, available for measurement, and with Q a known bound. n
Define a state error vector as
(3.1)
and a filtered error [16] as
, (3.2)
where is an appropriately chosen coefficient vector so that e(t) ? 0 exponentially as r(t) ? 0 (i.e. is Hurwitz). Then the dynamics (2.1) can be written in terms of the filtered error as
(3.3)
where . We will design an adaptive network controller using a subsequently defined adaptive law to keep r(t) bounded and small. If we can show that (3.2) is a stable system, this implies that e(t) remains bounded so that the tracking objective is achieved.
Definition 3. 1 : We say the solution of (3.2) is uniformly ultimately bounded (UUB) if for any U, a compact subset of ?n, and all x(t o) = xo? U there exists an e > 0 and a number T(e, xo) such that for all t 3 t o+T. n
For a general class of nonlinear systems defined below, an adaptive network controller is designed so that the output follows a prescribed trajectory with bounded error. Some system theory notions are given in this section. Any general smooth function can be expressed as [9]
, (3.4)
where W be the constant ideal network weights or parameters, is the basis function or regression matrix and is the reconstruction error vector whose bound is given by . For suitable one-layer NN approximation properties, must form a basis. For the case of NN approximators, it is well-known that radial basis functions (RBF), sigmoids, Bsplines form a basis. Here the function approximation is provided using one and two adaptive network scenarios.
3.1 Approximation-based Controller Using a Single Adaptive Network
A general sort of approximation-based controller is now derived by setting
, (3.5)
where an estimate of the function f(x) and being the PD outer tracking loop. Differentiating (3.2) and using (2.1), the closed-loop filtered error dynamics are found by
, (3.6)
where the function approximation error is given by .
Theorem 3.1: Let the desired trajectory for the combined system (base plus arm),, ,be bounded and the unknown disturbance forces be all zero. Let the control input for (2.1) be (3.5). Let the parameter tuning update is given by
, (3.7)
where being the constant adaptation gain matrix. The filetred tracking error, r(t), goes to zero with t and the parameter estimation error, , (or parameters, ) are bounded.
Proof: Under the ideal case, the error system is
(3.8)
and applying the LIP assumption one has
. (3.9)
The equation (3.9) can be rewritten as
(3.10)
Choose the Lyapunov function candidate
(3.11)
whose derivative is given by
. (3.12)
Evaluating the first derivative along the system trajectory (3.10) yields
(3.13)
Simplifying the first derivative to get,
. (3.14)
Now applying skew-symmetry property and substituting for with being constant from (3.7) into (3.14), equation (3.14) becomes
. (3.15)
Since V > 0 and 0, this shows stability in the sense of Lyapunov so that and (and hence ) are bounded. Thus
(3.16)
Now , and the boundedness of all the signals on the right-hand side of (3.9) verify the boundedness of and hence of , and therefore the uniform continuity of . This allows us to invoke Barbalat’s Lemma [9] in connection with (3.15) to conclude that goes to zero with t, and hence the tracking error . n
Remark : It can be shown that under an additional persistence of excitation (PE) condition, the parameter error goes to zero so that the parameter estimates are bounded. Here the unknown parameters for the MARS greenhouse application are the masses of the links, vehicle, unknown end-effector loads and friction coefficent. Since the PE condition is difficult to satisfy or to guarantee, the next theorem shows the UUB of the tracking error and the parameter estimates in the absence of PE condition [9].
Theorem 3.4: Let the desired trajectory for the vehicle,,,be bounded. Let the disturbance bound be , a known constant. Let the control input for (2.1) be (3.5). Let the parameter tuning update is given by
, (3.17)
where being the constant adaptation gain matrix and being a small design parameter. Let the robustifying signal v(t) be selected as
,
, Otherwise (3.18)
with a gain parameter and being a small positive number. Then the tracking error, e(t), is UUB and the parameter estimates are UUB.
Proof: Follow steps similar to Theorem 3.1. n
Remark: Theorems 3.1 and 3.2 present adaptation rules and stability analysis using an one adaptive network to approximate the nonlinear function f(.). Here one can employ a neural network, an adaptive controller and so on. For the case of adaptive control, the regression matrix forms a basis function and it has to be computed at every sampling interval. For the case of a mobile base plus arm, the regression matrix is very complex (see [5]) and computationally intensive. On the other hand, if we deploy a one layer neural network-based adaptive controller, there is no need to compute the regression matrix due to the basis functions; however the basis functions are typically nonlinear and computationally intensive. In the next section, we will use two one-layer adaptive networks to estimate the functions f(x) and g(x) to extend our approach to a broader class of nonlinear systems.
3.2 Adaptive NN System and Controller using two one-layer NN
In order to approximate f(x) and g(x) in (2.2), two NN are required. Therefore, from hereon the f(.) and g(.) can be written as
, (3.19)
, (3.20)
where are constant ideal weights and are functional reconstruction errors for f(.) and g(.) respectively. Since we have considered that the functions f(.) and g(.) in (2.1) are unknown, then one need to choose a control action
, (3.21)
where the estimates and will be constructed through NN, and the auxiliary term is
, (3.22)
with > 0. It is well-known, even in adaptive control of linear systems, that guaranteeing boundedness of away from zero becomes an important issue in this type of controller, as discussed next.
3.3 Proposed Controller
In order to guarantee the boundedness of away from zero for all well-defined values of x(k), , the control input is selected in terms of a control input, and a robustifying term, as
if I=1, if I = 0, (3.23)
where
, (3.24)
, (3.25)
the Indicator I in (3.23) is
I=1, If and
=0, Other wise (3.26)
with , , and s > 0 are design parameters. These modifications in the control input is necessary in order to ensure that the functional estimate is bounded away from zero. This results in a well-defined control everywhere and the uniformly ultimately boundedness of the closed-loop system can be shown by appropriately selecting the weight update algorithms.
4. Two-layer NN controller design
Here will be constructed by using an adaptive NN (which is a function approximator). Then be the current estimates. Note that with most adaptive approaches, one must find a basis function by a formal analysis of the system (2.1). That is, the basis vector depends on the unknown plant. By contrast, in NN control, the activation functions, , depends on the NN system design, and is good for any plant in the class (2.2), regardless of the specific forms of f(.) and g(.). By assumption the ideal values for exists and are bounded so that
, (4.1)
(4.2)
Define the weight estimation errors as
, (4.3)
and
. (4.4)
Let a NN is employed to approximate the continuous function f(x)
, (4.5)
with the weight tuning provided by
, (4.6)
a NN for g(x) as