gustavo freitas, fernando lizarralde, liu hsu, vitor

X SBAI – Simpósio Brasileiro de Automação Inteligente18 a 21 de setembro de 2011São João del-Rei - MG - Brasil

ISSN: 2175-8905 - Vol. X 977

A novel wheel-leg parallel mechanism control for Kinematic Reconfiguration of anEnvironmental Hybrid Robot

Gustavo Freitas, Fernando Lizarralde, Liu Hsu, Vitor Paranhos and Ney R. Salvi dos Reis

Abstract— This paper addresses a control design method for parallelmechanisms based on the kinematic constraints, rather than theconstraint equations. The method is applied on a new Environmentalrobot to control the position and angle of its wheels through 2DOFsuspensions with actuated parallel mechanisms. This case study isdiscussed based on the concepts of kinematic control and forward anddifferential kinematics. Experimental tests illustrate implementationissues and the efficacy of the proposed design.

I. INTRODUCTION

Mobile robots are widely employed in unstructured and haz-ardous terrains for several applications, including mining, forestry,agricultural and military activities ([1], [9]). For effective remoteoperation in inhospitable environments, efficient locomotion sys-tems, capable of working on irregular and rough terrains, must beconsidered.

This paper contemplates wheel-legged robots [8], which usewheels for propulsion and internal articulation to adapt their config-uration in order to accommodate for obstacles, also repositioningthe center of mass and influencing the contact forces against theterrain.

Accuracy, repeatability and payload capacities are fundamentalrequirements for the leg mechanism when advanced mobile robotsare considered. Parallel mechanisms provide a stiff connection be-tween the payload supported by the leg, with superior pose accuracythan serial chain mechanisms [10], [11]. The main disadvantagesrelated to this type of structures are related to workspace limita-tion, difficulties in mapping the forward kinematic and complexsingularity analysis [16], [13].

This paper presents a parallel mechanism control methodologybased on a strategy proposed in [16], where the differential kine-matic model is obtained considering the mechanism’s kinematicconstrains from its structure equations, instead of using explicitlythe constrains equations. Then, according to the system’s effectivedegrees–of–freedom (DOF), a control strategy with primary andauxiliary objectives is proposed considering the mechanism differ-ential kinematic model null space.

The described methodology is applied, controlling the suspen-sions of Environmental Hybrid Robot’s (EHR) new prototype [4].Each suspension is composed by a planar parallel mechanismwith 2 DOF, allowing the wheel’s position and orientation to beindependently controlled.

II. ENVIRONMENTAL HYBRID ROBOT

This work is motivated by the Environmental Hybrid Robot [5],[4] from Brazilian Oil company Petrobras S.A. (Fig. 1), which hasfour suspensions composed by parallel mechanisms. The robot ismeant to help monitoring the Amazon rain forest region, being usedas a maintenance tool for the Coari-Manaus pipeline [7].

G. Freitas, F. Lizarralde, L. Hsu and V. Paranhos are with the Dept. ofElectrical Eng., COPPE/Federal University of Rio de Janeiro, Brazil.

N. Reis are with CENPES / Petrobras S.A., Rio de Janeiro, Brazil.

This system is able to collect samples and environmental dataand also perform tasks in difficult-access areas of the forest. It isbeing designed to overcome obstacles and to operate on differentterrains, as firm ground with vegetation, sand, swamps, marshesand water surface with vegetation.

Fig. 1. Environmental Hybrid Robot’s new prototype with 2 DOFsuspensions.

An innovative locomotion system is designed according to theencountered conditions in Amazon. The wheel-legged architectureis adopted, allowing the robot to carry load and save energy, beingable to work on irregular terrains.

Each wheel is coupled to an independent suspension systemcomposed of a spring and electric linear actuators. The suspension’smotors are attached to screws, forming the active prismatic joints.

The suspensions’ parallel mechanism is designed for structuralrigidity, allowing the robot to overcome obstacles and increasingwheel’s traction.

The original suspension developed for the EHR [4] has only 1DOF. When the coupled motor of this structure is actuated, bothposition and angle of its wheel are amended. By commanding thewheel’s height, it is possible to modify the robot mobility, e.g. thesystem’s stability and force distribution among the legs [5], [4].Also, the suspension angle influences the contact point with theterrain and consequently the effective radio of the spherical wheels.Hence, the suspension’s reconfiguration affects the relation betweenvelocity and torque on each wheel.

The capability of influencing the system’s stability, its forcedistribution and also the relation between velocity and torque isdesirable. However, to optimize the robot’s missions, the suspen-sion mechanism should allow this parameters to be independentlycontrolled. Because of that, a new mechanism with 2 DOF is beingdeveloped, allowing the wheel’s height and angle to be decoupled.The new suspension also has a larger workspace, and even allowsthe robot to operate upside down, as presented in Fig. 2.

The robot uses 20” plastic bicycle wheels with solid tyres, whichwill be further covered by a profile as shown in Fig. 1 in order to

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Fig. 2. EHR’s 2 DOF suspensions in different configurations.

increase traction and allow the EHR to buoy. These floating wheelsare being designed to maximize traction considering the differentoperation environments.

The mobile robot weighs 35kg with 490mm of wheelbase andtrack width of 710mm. Nimh batteries supply energy for thesystem. A set of motors (70W for each wheel and 10W for eachsuspension active joint) and drivers (EPOs) from Maxon Motors areused, allowing the robot to navigate at firm terrains with maximumspeed of 1.50m/s.

The system is teleoperated. Communication with a base computeris accomplished through 900MHz Ethernet radios. The embeddedcontrols are executed by a PC/104 board with a 366MHz proces-sor. The robot’s orientation is obtained by an MTI Xsens inertialunit.

The new Environmental Robot prototype is still under develop-ment, and new equipments will be integrated into the system. Therobot’s careen and floating wheels, which are under construction,must be fixed before tests in the Amazon Rain Forest.

III. LEG FORWARD KINEMATICS

The forward kinematics for a robotic system is described bythe end-effector configuration specified for each chain. Here, theend-effector is represented by the contact point between wheel andterrain.

The contact point position is considered to be located in the wheelmost inferior point as presented in Fig. 3.

Each wheel is coupled to an independent suspension system with2 effective DOF (it can be verified using the Gluebler’s formula[12]), corresponding to an underactuated planar mechanism with 2closed kinematic chains.

The mechanism’s geometry is presented in Fig. 3. It is a 7 bar-link, where link 0 is fixed to the robot’s structure and the wheelcorresponds to the link segment 6. The leg has 2 active prismaticand 6 passive revolute joints.

Fig. 3. EHR’s suspension: seven-bar-link mechanism geometry withactive and passive joints.

In order to describe the leg’s forward kinematics, coordinateframes are attached to each link of it, where coordinate frame{xi, yi} is attached to the ith-link (see Fig. 4). The wheel-terraincontact position is defined by frame {xw, yw} and ~pij ∈ R3 isthe vector from the i-frame origin to j-frame origin represented inbase frame (frame 0). The mechanism’s links length are presentedin table I.

Fig. 4. EHR’s suspension: seven-bar-link mechanism frames.

Link # Parameter Value Unit

0 a0 185 mm

ψ0 117.0 deg

3 a30 80 mm

a31 40 mm

a32 40 mm

4 a4 70 mm

6 a60 75 mm

a61 115 mm

aw 275 mm

ψ6 112.0 deg

TABLE IEHR SUSPENSION MECHANISM LINKS LENGTH.

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The forward kinematics is described by the following structureequation [12]:

~p0w = ~p02 + ~p24 + ~p46′ + ~p6′w︸︷︷︸chain 1

= ~p02 + ~p24 + ~p46 + ~p6w︸︷︷︸chain 2

= ~p03 + ~p34 + ~p46′ + ~p6′w︸︷︷︸chain 3

(1)

θ0w = θ1 + θ2 + θ4 + θ6︸︷︷︸chain 1

= θ1 + θ2 + 2π + θ5 + π − ψ6︸︷︷︸chain 2

= ψ0 + θ3 + θ4 + θ6︸︷︷︸chain 3

(2)

where θ0w represents the orientation of the wheel frame with respectto base frame.

Equations (1)–(2) introduce constraints between the possible jointangles. These constraints allow the end-effector location controlto be done by specifying only the active joints variables θa =[d1, d2]T . The passive joints θp = [θ1,θ2,θ3,θ4,θ5,θ6]T take onvalues making equations (1)–(2) satisfied.

From (1), the forward kinematics is given by:

~p0w = ~p02 + ~p24 + ~p46′ + ~p6′w

= d2 R01(θ1) x+ (a30 + a31) R01(θ1)R23(θ2) x

+a4 R01(θ1)R23(θ2)R34(θ4) x

+a61 R01(θ1)R23(θ2)R34(θ4)R46′(θ6) x

−aw R01(θ1)R23(θ2)R34(θ4)R46′(θ6) y (3)

where a30, a31, a4, a61, aw are values from table I, x = [1, 0, 0]T ,y = [0, 1, 0]T and the elementary rotation matrix Rij(θi) ∈SO(3) is the orientation of j-frame with respect to i-frame, givenby:

Rij(θi) =

cos(θi) − sin(θi) 0sin(θi) cos(θi) 0

0 0 1

(4)

Due to the parallel mechanism’s structure constraints, it is nottrivial to obtain an analytical solution for the forward kinematicsin terms of θa, i.e. ~p0w = k1(d1, d2), θ0w = k2(d1, d2).

The passive revolute joints θp ∈ R6 are obtained in termsof the active joints d1, d2, i.e. θ1 = f1(d1), θ3 = f3(d1),θ4 = f4(d1), θ5 = f5(d2), θ6 = f6(d2). These functions canbe calculated through algebraic identities and geometry (by usingcosine formula).

Another possibility to determine θp is to reduce the forwardkinematic problem into appropriate subproblems whose solutionsare known [12]. The passive joints θp can be calculated throughPaden-Kahan subproblems 1 and 3 [12, pag. 99–103], which aregeometrically meaningful and numerically stable.

Knowing θp, it is possible to obtain the contact point position~p0w through equation (3) and orientation (θ0w) using (2).

Figure 5 illustrates suspension’s configurations for d1 ∈[170, 220]mm and d2 ∈ [45, 105]mm. The mechanism’s dexterousworkspace in terms of p0w and θ0w is presented in Fig. 6.

IV. DIFFERENTIAL KINEMATICS

The planar parallel mechanism end-effector’s (wheel-terrain con-tact point) velocity is related to the joints’ velocities by differen-tiating the structure equation (1). This gives a Jacobian matrix for

Fig. 5. EHR’s mechanism’s configurations in terms of θa.

Fig. 6. EHR’s suspension’s dextrous workspace.

each chain:

vw = S J1

θ1d1θ2θ4θ6

= S J2

θ1d1θ2d2θ5

= S J3

θ3θ4θ6

(5)

where vw = [p0wx , p0wy , ω0w]T ∈ R3 is the planar linear velocityin x0, y0 axes and angular velocity (ω0w = θ0w), S ∈ R3×6 is aselection matrix considering the planar mechanism:

S =

1 0 0 0 0 00 1 0 0 0 00 0 0 0 0 1

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and the 6-DOF Jacobian matrices are given by

J1 =

[z × ~p0w R01x z × ~p2w z × ~p4w z × ~p6′w

z 0 z z z

]J2 =

[z × ~p0w R01x z × ~p2w R01R23x z × ~p6w

z 0 z 0 z

]J3 =

[z × ~p3w z × ~p4w z × ~p6′w

z z z

]where z = [0, 0, 1]T , ~p0w is given in (3) and

~p2w = (a30 + a31) R01(θ1)R23(θ2) x+ ~p4w (6)~p3w = a31 R(ψ0)R03(θ3) x+ ~p4w (7)~p4w = a4 R01(θ1)R23(θ2)R34(θ4) x+ ~p6′w (8)~p6w = a60 R01(θ1)R23(θ2)R36(θ5) p6w (9)p6w = [a60 cosψ6 + a61; −a60 sinψ6 − aw; 0]T (10)~p6′w = a61 R01(θ1)R23(θ2)R34(θ4)R46′ (θ6) x

−aw R01(θ1)R23(θ2)R34(θ4)R46′ (θ6) y (11)

The Jacobian can be written in a more conventional form bystacking the Jacobians for each chain: S J1 0 0

0 S J2 00 0 S J3

︸︷︷︸

Jext

θext =

III

︸︷︷︸

A

vw (12)

or equivalently:Jext θext = A vw (13)

where matrix A ∈ R9×3 is always full column rank, Jext ∈ R9×13

and θext = [θ1, d1, θ2, θ4, θ6, θ1, d1, θ2, d2, θ5, θ3, θ4, θ6]T .It is possible to observe that joints θ1, d1, θ2, θ4, θ6 are duplicated

in equation (12), and because of that, Jext has more columns thannecessary. Defining θext = M θ, with

M =

I3 03×2 03×3

02×3 I2 02×3

I3 03×2 03×3

03×3 03×2 I3

02×3 I2 02×3

,where Ii is an identity matrix and 0i×j is a zero matrix with i rowsand j columns, equation (12) is rewritten as:

J θ = A vw (14)

with J ∈ R9×8 = Jext M and θ = [θ1, d1, θ2, θ4, θ6, d2, θ5,θ3]T .

Considering A† ∈ R3×9 the pseudo-inverse of A such thatA†A = I , and A ∈ R6×9 the annihilator matrix as AA = 0, thedifferential kinematic relationship (14) can be equivalently writtenas [17], [14]:

vw = Jt θ, Jc θ = 0 (15)

where Jt = A†J and Jc = AJ .Considering the active joints [d1, d2]T = θa and collecting all

the passive joints’ velocities together in θp and then partitioningJt ∈ R3×8 and Jc ∈ R6×8 accordingly

Jt = [Jta ; Jtp ], Jc = [Jca ; Jcp ] (16)

considering (15) and (16), one has that:

vw = Jtaθa + Jtp θp, Jca θa + Jcp θp = 0 (17)

where Jta∈R3×2, Jtp∈R3×6, Jca∈R6×2 and Jcp∈R6×6.

If Jcp is full rank, the passive joints’ velocities can be given interms of active joint velocity θa:

θp = −J−1cp Jca θa (18)

It is possible to verify that Jcp is not singular for the consideredmechanism.

Finally, the wheel velocity vw is given in terms of the activejoints’ velocities d1, d2 by:

vw = (Jta − Jtp J−1cp Jca)︸︷︷︸

Jp

θa (19)

where Jp ∈ R3×2 is the leg’s Jacobian.

V. KINEMATIC CONTROL

A kinematic control approach is proposed to change the suspen-sion’s parallel mechanism posture in order to accomplish an specifictask. Considering the system with 2 effective DOF, the adoptedcontrol has two objectives:

1) control the vertical distance p0wy between wheel-terraincontact point and the suspension frame:

p0wy → p∗0wy(t) eh = p∗0wy

− p0wy → 0 (20)

2) regulate the wheel’s orientation θ0w actuating on the mech-anism null space velocities with respect to the primaryobjective:

θ0w → θ∗0w eo = θ∗0w − θ0w → 0 (21)

The proposed scheme consists in commanding the robot’s joints’velocities d1, d2, i.e. u(t) = θa, adjusting the contact point’s poseto cancel the errors (20) and (21).

The actuated planar suspension’s differential kinematic model(19) can be decoupled in 2 components: a linear p0wy and anangular velocity ω0w as:[

p0wy

θ0w

]= Jp θa; Jp =

[JpyJθ

]where Jpy , Jθ ∈ R1×2, thus the joints’ velocities can be obtainedfrom θa= Jp

−1v, where v is a cartesian control law.

From Fig. 6, it is possible to observe that the suspension’sworkspace has boundary singularities in terms of orientation. Con-sidering p0wy < −320mm, the orientation singularity is achievedfor θ∗0w >= 6π rad.

However, in these configurations, it is still possible to controlp0wy . The vertical distance between suspension frame and wheel-terrain contact point directly influences the robot’s orientation, forcedistribution among legs and tip over stability as presented in [4],[3]. Thus it corresponds to the main control objective, avoidingorientation singularities.

The adopted approach consists on given priority to control thevertical distance, considering orientation as an auxiliary control.Therefore, the mechanism is considered redundant, with 1 effectiveDOF and 2 active joints. The differential kinematic is given byp0wy = Jpy θa, and the joints’ velocities are given by [15], [2]:

θa = J†py v + (I − J†py Jpy )︸︷︷︸P

θa0 (22)

where col(P) spans the null space of Jpy , and θa0 is an arbitraryvector of active joints’ velocities. The right hand side of (22)can be interpreted as null space velocity which provides internalmovements with respect to the redundant mechanism’s model.

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Then, using a proportional law as

u = J†pyKh eh + (I − J†pyJpy ) u , (23)

where u is the auxiliary control signal, the position error dynamicis determined by eh+Kh eh = 0, since the right hand side of (23)spans the null space of Jpy . Considering Kh > 0 and nonsingularJpy , we have limt→∞ eh(t) = 0.

The auxiliary control u is chosen to improve the mechanism’sperformance during tasks execution. A typical choice is

u = K

(∂f(θa)

∂θa

)T, (24)

where K>0 is a gain factor and f(θa) is an objective function1 interms of θa, here chosen to cancel the orientation error: f(θa) = e2o.

Considering f(θa) = e2o, the auxiliary control is given by:

u = Ko eo eo = Ko JTθ eo (25)

where Ko > 0 is the orientation gain.Finally, the active joints’ velocities θa are defined as:

u = θa = J†pyKh eh + (I − J†pyJpy ) Ko JTθ eo (26)

VI. CONTROL IMPLEMENTATION AND EXPERIMENTAL

RESULTS

This section describes the kinematic control’s implementation inthe Environmental Robot. The control algorithm is executed on aPC/104 single board computer featuring a low-power, 366 MHzAMD Geode GX500 processor running Linux (Debian 5.0.1, Kernel2.6.34), which is embedded in the robot.

To achieve real-time code execution, Xenomai 2.5.3 is usedside-by-side with Linux [6]. Xenomai uses the Adaptive DomainEnvironment for Operating Systems (ADEOS) to share hardwareinterrupts with Linux and guarantee real-time operation. The im-plementation described here uses Xenomai’s native API.

The EHR’s software is implemented in C/C++. The user-spaceapplication spawns the following real-time task that executes thecontrol algorithm:

openLogFiles();startCommunicationWithHardware();setControlParameters();while(true){

for(allMechanisms){readActualPositions();computeForwardKinematics();getReferences();calculateErrors();computeVelocities();

}for(allMechanisms){

sendHardwareTheVelocities();logValues();

}}

Among the several custom functions that comprise the code, thefollowing two are particularly relevant:

• forwardKinematics(): Calculates the leg’s forward kinematics~p0w, θ0w from the active joint position vector qa.

• suspensionControl(): Calculates the control command u = qafrom the joint position vectors qa, qp and closed-loop positionand orientation errors eh, eo.

1f(·) is continuous, differentiable and convex.

The function forwardKinematics is obtained with Matlab’s ccodecommand. The mathematical expression for the forward leg kine-matics is declared as a symbolic expression in Matlab, and ccode iscalled to generate the C code implementing the expression. Becauseccode does not always generate computationally-efficient code, wehand-customized the code to reduce computation speed, by storingin memory those variables that are calculated frequently and reusingthem during code execution. This manual optimization step resultedin a code that runs 80% faster (see Table II).

The suspensionControl function is also obtained via ccode andmanual optimization. The matrix inversion J−1

cp is computed usingthe open source computer vision library named OpenCV. Thelibrary provides functions for inverting matrices using singularvalue decomposition (SVD) as well as Gaussian elimination withoptimal pivot element selection (LU). The computational effort forboth the original and optimized codes using each of the matrixinversion methods is shown in Table III. The optimized code withLU inversion runs 38% faster than its original counterpart.

Algorithm E[t] σ

Non-Optimized forwardKinematics 165.79µs 74.72µs

Optimized forwardKinematics 35.19µs 6.50µs

TABLE IIforwardKinematics COMPUTING MEAN TIME E[T] AND ASSOCIATED

STANDARD DEVIATION σ.

Algorithm E[t] σ

Non-Optimized with LU inversion 353.60µs 99.79µs

Non-Optimized with SVD inversion 1074µs 158.93µs

Optimized with LU inversion 219.07µs 65.05µs

Optimized with SVD inversion 942.57µs 175.61µs

TABLE IIIsuspensionControl COMPUTING MEAN TIME E[T] AND ASSOCIATED

STANDARD DEVIATION σ.

In the experiments reported here the wheel’s vertical positionreference p∗0wy

is a series of step signals, and the orientationreference is such that the wheel is perpendicular to the ground.The results are presented in Fig. 7. It can be seen that the primarycontrol objective of the vertical position of the wheel-terrain contactpoint is accomplished, with eh → 0 within a time of about 3 s.

Likewise, the secondary control objective is also achieved, withthe orientation error eo tending to zero at every change in referenceposition. Because of the reduced priority placed on orientationcontrol, the time constant is longer, on the order of 20 s. Notethat the orientation error is not canceled when the active jointd1 reaches its minimum position at t = [140, 160] s. This,however, is a physical limitation of the leg’s mechanism and notan intrinsic limitation of the controller. More importantly, even atthese configurations the null-space control methodology still allowsfor full control of the vertical distance p0wy , since the orientationcontrol acts on the null space of Jpy and therefore does not interferewith the control of p0wy .

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Fig. 7. EHR’s suspension kinematic control experimental results.

VII. CONCLUSIONS AND FUTURE WORKS

This paper presents a control methodology for parallel mecha-nisms, based on a modeling procedure proposed in literature whichconsider the mechanism’s kinematic constraints from its structureequations, instead of explicitly using the constraints equations. Akinematic control is then proposed considering the mechanism’sDOF to improve achieving primary and auxiliary objectives.

The presented methodology is demonstrated considering theparallel 2 DOF suspension mechanism of the Environmental HybridRobot’s new prototype. The mechanism’s forward and differentialkinematics are obtained. Then, a strategy is presented to controlfirstly the vertical distance between wheel and base frame and thenthe wheel’s orientation.

Experimental results are presented considering a constant ori-entation reference θ∗0w and a sequence of steps as vertical positionreference p∗0wy

. The control implementation is described, presentingdetails related to real time computation. Tests with the EHR are alsopresented.

Future works will consist on commanding the combined sus-pensions in order to improve the robot mobility, i.e. traction andstability, as proposed in [4], [3].

The Environmental Hybrid Robot’s new prototype is still underdevelopment. Some preliminary tests have been accomplished inlaboratory and field, and are presented in Fig. 8. The robot’scomplete functionality will be achieved when the floating wheelsare integrated into the system.

VIII. ACKNOWLEDGMENTS

The authors would like to thank all the Petrobras S.A. RoboticsLaboratory team.

This work was partially supported by FAPERJ, CNPq, CAPESand Petrobras S.A..

Fig. 8. EHR during preliminary field tests.

REFERENCES

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[3] G. Freitas, G. Gleizer, F. Lizarralde, and L. Hsu, “Multi-objectiveoptimization for kinematic reconfiguration of mobile robots,” in Proc.of the 6th annual IEEE Conf. on Automation Science and Eng.,Toronto, August 2010, pp. 686–691.

[4] G. Freitas, G. Gleizer, F. Lizarralde, L. Hsu, and N. R. S. dos Reis,“Kinematic reconfigurability control for an environmental mobile robotoperating in the amazon rain forest,” Journal of Field Robotics, vol. 27,no. 2, pp. 197–216, March–April 2010.

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[14] ——, “Determination of unstable singularities in parallel robots withN-arms,” IEEE Transactions on Robotics, vol. 22, no. 1, pp. 160–167,February 2006.

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[16] J. T. Wen and J. F. O’Brien, “Singularities in three-legged platform-type parallel mechanisms,” IEEE Transactions on Robotics and Au-tomation, vol. 19, no. 4, pp. 720–727, 2003.

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