Why we teach what we teach is just as important as why we study what we study but is seldom discussed as a field-defining issue. *American Literature’s* most recent special issue, “Pedagogy: Critical Practices for a Changing World,” edited by Carol Batker, Eden Osucha, and Augusta Rohrbach, integrates discipline-specific knowledge more fully into a critical discussion of pedagogy. By leveraging the location of pedagogy as developing out of specific scholarly concerns, articles within this issue illustrate the intersection of theory and pedagogical practice while highlighting the diverse disciplinary, institutional, and political contributions of American literature to higher education and community-based teaching and learning.

In turning their attention to pedagogy, the editors of this special issue ask both how scholarly engagement with American literature has produced a distinct set of pedagogical practices and how pedagogical practices raise new questions about the relevance and role of American literature. Rather than focusing on a particular teaching strategy or text, these essays approach the topic from larger philosophical and disciplinary perspectives.

Read the special editors’ introduction to the issue, made freely available now through August 26, 2017.

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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

Abstract

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.

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) * * exponentially as r(t) * * (i.e. is Hurwitz). Then the dynamics (2.1) can be written in terms of the filtered error as

(3.3)

where Simulations have been conducted by employing the dynamic equations of the combined mobile robot plus arm system for both maintaining a desired course and speed. A typical case study of the end-effector extension while the base is in motion is illustrated during trajectory following. Here a conventional adaptive controller approach is pursued where the mass of the vehicle and the links along with the vehicle friction is considered unknown. The regression matrix is given in [7]. The following parameters are used in the simulation. mv =10=mass of the mobile base; m4=5kg=mass of the first link of the manipulator; m5=5kg=mass of the second link of the manipulator; m6=3kg=mass of the third link of the manipulator; endeffector load effect for the third link = 2 kg; k=1kg/s =coefficient of friction for the base on the ground g=9.8 kgm/s2 ; l5=0.5m=length of the second link; l6=0.5m= length of the third link; kp=200=derivative gain; kv=40=proportional gain; kv is chosen as a function of kp to obatin critically damped response, ; Arm inertia: Iz1=0.84kgm2; Iz4=0.0427 kgm2; Ix5=0.1302 kgm2; Iz5=0.0427 kgm2; Ix6=0.1302 kgm2; Iz6=0.0427 kgm2; Iy5=0.1208 kgm2; Iy6=0.1208 kgm2; r=0.25m=radius of the wheel; R=1m=with of the mobile base.A cubic spline algorithm is used to plan a path and the desired trajectory is specified as follows. links and the friction coefficient, and unknown endeffector loads, forces the base plus arm to follow the desired trajectory. On the other hand, the PD controller shown in Figures 6.3 and 6.4, though renders bounded errors, it is not suitable for MARS greenhouse operation tasks.

7. Conclusions

An adaptive framework is introduced for the control of a mobile base with an onboard arm. In this adaptive framework, a conventional adaptive controller and all other functional approximators can be used. Stability analysis is given using the Lyapunov criteria for this adaptive network controller. It is shown that the closed-loop system is stable with bounded disturbances. Under ideal conditions, when no disturbances are present, parameters are bounded and the tracking error is shown to converge to zero. When disturbances are present, UUB of both tracking error and parameter estimates are given.

Due to an inner adaptive loop, the dynamics due to the motion of the onboard arm is automatically compensated. The net result is an adaptive system, which will allow coordinated motion of the base and the arm. Simulation results show that the arm motion, which acts as a disturbance, is automatically compensated by the controller. From the results, an adaptive network controller outperforms a PD controller and is recommended for our MARS greenhouse operations. Future work will involve extending this adaptive network controller for other tasks including force control operations.

8. References

[1] Y. Yamamoto, and X. Yun, “Coordinated obstacle avoidance of a mobile manipulator”, Proceedings of the IEEE Conference on Robotics and Automation, pp.2255-2260, 1995.

[2] Y. Yamamoto, and X. Yun, “Unified analysis on mobility and manipulability of mobile manipulators”, Proceedings of the IEEE Conference on Robotics and Automation, pp.1200-1206, 1999.

[3] R. Colbaugh, “Adaptive stabilization of mobile manipulators”, Proc. of the Amer. Controls Conf., pp. 1-5, 1998.

[4] A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization of nonholonomic caplygin dynamic systems”, Proc. of the IEEE Conference on Decision and Control, using (3.5), the error system for trajectory following is represented in the Brunosky canonical form [5] as

, (5.2)

with A,B given in equation (3.5), and the states of the error system given by where

. (5.3)

In the case of prescribing a course and speed, the error system is (5.2), with A and B given by equation (2.5a) and where the desired course and speed are , and the desired acceleration, is considered to be zero. The controller should yield suitable stability properties for both cases.

6. Results

; ; .

The mobile base was requested to move with a linear velocity of 10 units and a heading angle of radians. During the motion, at t = 30 units the three degrees-of-freedom (dof) of the on-board arm were allowed to move from their current positions with a tray that has been picked. The results corraborate that the mobile base traveling at the desired speed attains the desired heading angle(see Fig. 6.1 and 6.2) while the interactive forces/disturbances introduced were compensated. These results show that the adaptive controller while estimating the masses of the vehicle, the

pp. 1127-1132, December 1991.

[5] S. Jagannathan, F. L. Lewis and K. Liu, “Motion control and obstacle avoidance of mobile robot with an onboard manipulator”, Journal of Intelligent Manufacturing Systems, vol.5, pp. 287-302, 1994.

[6] S. Jagannathan, S. Q. Zhu and F. L. Lewis, “Path planning and control of a mobile base with nonholonomic constraints”, Robotica, vol. 12, part 6, pp. 529-540, 1994.

[7] S. Jagannathan and A. Levesque, “An Adaptive Network Framework for Control”, Tech Report, Dept. of Electrical Engineering, The University of Texas at San Antonio, June 2000.

[8] I. Kanellakapoulos, P.V. Kokotovic, and A.S.Morse, “Systematic design of adaptive controllers for feedback linearizable nonlinear systems”, IEEE Trans. on Auto. Control, vol.35, no.4, pp.416-424, 1994.

[9] A. Yesilderik and F. L. Lewis, “Feedback linearization using neural networks”, Automatica, pp. 1659-1664, vol.31, no.11, November, 1995.

Fig. 6.1: Adaptive controller: a) Heading and Joint angles (b)

Link 4 (c) Link 5 (c) Link 6.

Fig. 6.2: Response of the adaptive controller: g) Error,

h) Unknown Masses and i) Friction.

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an other aspects is this as say prof dr mircea orasanu and prof horia orasanu as followings

In order to perform control design, we need to write the governing equations of the system in an implementable way. This is often done by choosing a local coordinate chart and writing the system equations with respect to (w.r.t.) these coordinates. According to the illustration of the snake robot in Figure Figure1,1, we choose the vector of the generalized coordinates of the N-link snake robot as x = [q1,q2,…,qN−1,θN,px,py]T ∈ ℝN+2, where qi with i∈{1,…,N−1} denotes the ith joint angle, θN denotes the head angle, and the pair (px,py) describes the position of the CM of the robot w.r.t. the global x−y axes. Since the robot is not subject to nonholonomic velocity constraints, the vector of the generalized velocities is defined as x˙=[q˙1,q˙2,…,q˙N−1,θ˙N,p˙x,p˙y]T∈RN+2. Using these coordinates, it is possible to specify the kinematic map of the robot. In this paper, we denote the first N elements of the vector x, i.e. (q1,…,qN−1,θN), as the angular coordinates, and the corresponding dynamics as the angular dynamics of the system.

Figure 1

An illustration of the N -link snake robot. Kinematic parameters of the snake robot.

This is due to the fact that it is usually not straightforward to integrate the anisotropic external dissipative forces, i.e. ground friction forces, acting on these complex robotic structures into their Euler-Lagrange equations of motion. However, ground friction forces have been proved to play a fundamental role in snake robot locomotion (see, e.g. [16]). In this paper, we derive the equations of motion of the snake robot in a Lagrangian framework, i.e. treating the robot as a whole and performing the analysis using a Lagrangian function, which is simple to follow and better suited for studying advanced mechanical phenomena such as elastic link deformations [25], which might be insightful for future research challenges on snake robots. Moreover, we integrate the anisotropic friction forces into these equations using the Jacobian matrices of the links, which gives a straightforward mapping of these forces for the equations of motion.

Snake robots are a class of simple mechanical systems, where the Lagrangian L(qa,x˙) is defined as the difference between the kinetic energy K(qa,x˙) and potential energy 𝒫(x) of the system [26]. Since the planar snake robot is not subject to any potential field, i.e. −∇𝒫(x) = 0, we may write the Lagrangian equal to the kinetic energy, which is the sum of the translational and the rotational kinetic energy of the robot:

L(qa,x˙)=K(qa,x˙)=12m∑i=1N(p˙2x,i+p˙2y,i)+12J∑i=1Nθ˙2i

(9)

where m and J denote the uniformly distributed mass and moment of inertia of the links, respectively. Using the Lagrangian function (9), we write the Euler-Lagrange equations of motion of the controlled system as

ddt[∂L(qa,x˙)∂x˙i]−∂L(qa,x˙)∂xi=(B(x)τ−τf)i

(10)

where i∈{1,…,N+2}, B(x) = [ej] ∈ ℝ(N+2)×(N−1) is the full column rank actuator configuration matrix, where ej denotes the jth standard basis vector in ℝN+2. Moreover, B(x)τ ∈ ℝN+2 with τ = [τ1,…,τN−1]T ∈ ℝN−1 stands for the generalized forces resulting from the control inputs. Furthermore, τf=[τ1f,…,τN+2f]T∈RN+2 denotes viscous and Coulomb friction forces acting on (N+2) DOF of the system. The controlled Euler-Lagrange equations (10) can also be written in the form of a second-order differential equation as

M(qa)x¨+C(x,x˙)x˙=B(x)τ−τf

(11)

where M(qa) ∈ ℝ(N+2)×(N+2) is the positive definite symmetric inertia matrix, C(x,x˙)x˙∈RN+2 denotes the generalized Coriolis and centripetal forces, and the right-hand side terms denote the external forces acting on the system. The fact that the inertia matrix is only a function of the directly actuated shape variables qa is a direct consequence of the invariance of the Lagrangian

The geometry of the problem

The (N+2)-dimensional configuration space of the snake robot is denoted as 𝒬 = 𝒮 × 𝒢, which is composed of the shape space and a Lie group which is freely and properly acting on the configuration space. In particular, the shape variables, i.e. qa=(q1,…,qN−1), which define the internal configuration of the robot and which we have direct control on, take values in . Moreover, the position variables, i.e. qu=(θN,px,py), which are passive DOF of the system, lie in . The velocity space of the robot is defined as the differentiable (2N+4)-dimensional tangent bundle of as T𝒬 = 𝕋N × ℝN+4, where 𝕋N denotes the N-torus in which the angular coordinates live. The free Lagrangian function of the robot ℒ:T𝒬 → ℝ is invariant under the given action of on . The coupling between the shape and the position variables causes the net displacement of the position variables, according to the cyclic motion of the shape variables, i.e. the gait pattern. Note that for simplicity of presentation, in this paper, we consider local representation of T𝒬 embedded in an (2N+4)-dimensional open subset of the Euclidean space ℝ2N+4. To this end, we separate the dynamic equations of the robot given by (11) into two subsets by taking x = [qa,qu]T ∈ ℝN+2, with qa ∈ ℝN−1 and qu ∈ ℝ3 which were defined in the subsection describing the geometry of the problem:

m11(qa)q¨a+m12(qa)q¨u+h1(x,x˙)=ψ∈RN−1

(20)

m21(qa)q¨a+m22(qa)q¨u+h2(x,x˙)=03×1∈R3

(21)

where m11 ∈ ℝ(N−1)×(N−1), m12 ∈ ℝ(N−1)×3, m21 ∈ ℝ3×(N−1), and m22 ∈ ℝ3×3 denote the corresponding submatrices of the inertia matrix, and 03×1 = [0,0,0]T ∈ ℝ3. Furthermore, h1(x,x˙)∈RN−1 and h2(x,x˙)∈R3 include all the contributions of the Coriolis, centripetal, and friction forces. Moreover, ψ ∈ ℝN−1 denotes the non-zero part of the vector of control forces, i.e. B(x)τ = [ψ,03×1]T ∈ ℝN+2. From (21), we have

q¨u=−m−122(h2+m21q¨a)∈R3

(22)

Substituting (22) into (20) yields

(m11−m12m−122m21)q¨a−(m12m−122)h2+h1=ψ

(23)

For linearizing the dynamics of the directly actuated DOF, we apply the global transformation of the vector of control inputs as

ψ=(m11−m12m−122m21)ϑ−(m12m−122)h2+h1

(24)

where 𝜗 = [𝜗1,𝜗2,…,𝜗N−1]T ∈ ℝN−1 is the vector of new control inputs. Consequently, the dynamic model (20)-(21) can be written in the following partially feedback linearized form

q¨a=ϑ∈RN−1

(25)

q¨u=D(x,x˙)+C(qa)ϑ∈R3

(26)

with

D(x,x˙)=−m−122(qa)h2(x,x˙)=[fθN,fx,fy]T∈R3

(27)

C(qa)==−m−122(qa)m21(qa)[βi(qa),0,0]T∈R3×(N−1)

(28)

where βi(qa):𝒬 → ℝ is a smooth scalar-valued function. It can be numerically shown that the value of βi is negative at any configuration qa ∈ 𝒬. Furthermore, fθN, fx, and fy denote the friction forces acting on θN, px, and py, respectively ( fθN also contains Coriolis forces besides the friction forces). For the aim of analysis and model-based control design, we write (25)-(26) in a more detailed form:

q¨a=ϑ∈RN−1

(29)

θ¨N=fθN(x,x˙)+βi(qa)ϑi∈R

(30)

p¨x=fx(x,x˙)∈R

(31)

p¨y=fy(x,x˙)∈R

(32)

where the summation convention is applied in (30), and henceforth, to all the equations which contain repeated upper-lower indices (i.e. whenever an expression contains a repeated index, one as a subscript and the other as a superscript, summation is implied over this index [26]). The dynamical system (29)-(32) is in the form of a control-affine system with drift. In particular, the term

A(x,x˙)=[q˙a,q˙u,0(N−1)×1,D(x,x˙)]T∈R2N+4

(33)

is called the drift vector field, which specifies the dynamics of the robot when the control input is zero. Furthermore, the columns of the matrix

B(qa)=⎡⎣⎢⎢⎢⎢0(N+2)×(N−1)IN−1[β1(qa),…,βN−1(qa)]02×(N−1)⎤⎦⎥⎥⎥⎥∈R(2N+4)×(N−1)

(34)

are called the control vector fields, which enable us to control the internal configuration and consequently the orientation and the position of the robot in the plane.

Remark2.

The last two rows of the control vectors in (34) are composed of zero elements. This implies that the control forces have no direct effect on the dynamics of the position of the CM of the robot, i.e. (31)-(32). Furthermore, the dynamics of the position of the CM are coupled with the dynamics of the directly actuated shape variables qa, i.e. (29), only through the friction forces. Accordingly, in the absence of the friction forces, the linear momentum of the robot is a conserved quantity, and the position of the CM of the robot is not controllableTo define the guidance law, without loss of generality, we assign the global coordinate system such that the global x-axis is aligned with the desired path. Consequently, the position of the CM of the robot along the y-axis, denoted by py, defines the shortest distance between the robot and the desired path, often referred to as the cross-track error. In order to solve the path following problem, we use the LOS guidance law as a virtual holonomic constraint, which defines the desired head angle as a function of the cross-track error as

ΦN=−tan−1(pyΔ)

(46)

where Δ>0 is a design parameter known as the look-ahead distance. The idea is that steering the head angle of the snake robot such that it is headed towards a point located at a distance Δ ahead of the robot along the desired path will make the snake robot move towards the path and follow it.

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GEOMETRIC ASPECT AND CURVATURE FOR NONHOLONOMIC PROBLEM

Author Horia Orasanu

ABSTRACT

In the Cartesian coordinate system, all three basis vectors are absolute constants:

The derivative of a vector is then straightforward to calculate:

But many non-Cartesian basis vectors are not constant.

Cylindrical Polar:

Penalty functions are not effective if the optimum lies in the boundary between the feasible and the infeasible regions or when the feasible region is disjoint

1 INTRODUCTION

Researchers have also proposed a number of other approaches to handle constraints such as the self-adaptive penalty, epsilon constraint handling and stochastic ranking. Developing novel constraint handling methods and investigating the performances of search engines on solving constrained problems have attracted much interest recently.

Irrespective of the high level of interest in constrained real-parameter optimization, the current constrained optimization test suite (CEC 2006 Benchmark Problems) has dimensions between 2 and 20 which is very low. In addition, CEC 2006 benchmark has been solved satisfactorily by several methods. Therefore, it has become impossible to demonstrate the superior performance of newly designed algorithms. Therefore, there is an urgent need to upgrade the current test suite by increasing dimensional scalability and by considering the types of constraints (equality, inequality, linear, nonlinear, dimensionality, active, etc.), types of objective functions (linear, quadratic, nonlinear, multimodality, separability, etc.), connectivity / relative size of feasible region and so on.

Given that the gradient operator in a general curvilinear coordinate system is

, why isn’t the divergence of

equal, in general, to

The quick answer is that the differential operators operate not just on the components , but also on the basis vectors . In most orthonormal coordinate systems, these basis vectors are not constant. The divergence therefore contains additional terms. =

For Cartesian coordinates, all derivatives of any basis vector are zero, which leaves the familiar Cartesian expression for the divergence. But for most non-Cartesian coordinate systems, at least some of these partial derivatives are In spherical polar coordinates, naming the three basis vectors as we have:

The relationship to the Cartesian coordinate system is

.One of the scale factors isIn a similar way, we can confirm that .

All of the above are undefined on the z-axis (sin = 0), where there is a coordinate singularity. However, by taking the limit as , we may obtain well-defined values for some or all of the above expressions.

not zero. More complicated expressions for the divergence therefore arise.

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we consider here some aspects as say prof dr mircea orasanu and prof horia orasanu as followings

VIRTUAL VARIATIONS FOR NONHOLONOMIC CONSTRAINTS

Author Horia Orasanu

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