A transverse local feedback linearization approach to human head movement control

Takafumi OKI,Bijoy K.GHOSH

Department of Mathematics and Statistics,Texas Tech University,Lubbock,Texas 79409-1042,U.S.A.

A transverse local feedback linearization approach to human head movement control

Takafumi OKI,Bijoy K.GHOSH?

Department of Mathematics and Statistics,Texas Tech University,Lubbock,Texas 79409-1042,U.S.A.

In the mid-nineteenth century,Donders had proposed that for every human head rotating away from the primary pointing direction,the rotational vectors in the direction of the corresponding axes of rotation,is restricted to lie on a surface.Donders’intuition was that under such a restriction,the head orientation would be a function of its pointing direction.In this paper,we revisit Donders’Law and show that indeed the proposed intuition is true for a restricted class of head-orientations satisfying a class of quadratic Donders’surfaces,if the head points to a suitable neighborhood of the frontal pointing direction.Moreover,on a suitably chosen subspace of the 3D rotation group SO(3),we describe a head movement dynamical system with input control signals that are the three external torques on the head provided by muscles.Three output signals are also suitably chosen as follows.Two of the output signals are coordinates of the frontal pointing direction.The third signal measures deviation of the state vector from the Donders’surface.We claim that the square system is locally feedback linearizable on the subspace chosen,and the linear dynamics is decomposed into parts,transverse and tangential to the Donders’surface.We demonstrate our approach by synthesizing a tracking and path-following controller.Additionally,for different choices of the Donders’surface parameters,head gaits are visualized by simulating different movement patterns of the head-top vector,as the head-pointing vector rotates around a circle.

Head movement,Donders’surface,transverse feedback linearization,Fick-Gimbal

1 Introduction

The subject matter of this paper is to control the pointing direction of a rigid body,such as the human eye or the head(see Robinson[1]and Peng et.al.[2]).The control inputs are generated by external torques from a set of muscles rotating the rigid body.The goal is to make the“pointing direction”(for example,the nose direction of the head)of the body directed towards a point target.If this target is moving along a path already predefined in R3,then the“body pointing direction”needs to track the direction of the point target,in order to keep the “target in view”.Even after pointing towards a target,the orientation of the eye/head being controlled remains ambiguous and the “head movement controller”disambiguates the orientation by imposing additional constraint on the axes of rotation.Such constraints were originally proposed(for both the eye and the head movement problems)in 1845 by physiologists such as Listing,Donders and Helmholtz,among others[3].In the case of head movement,this constraint is popularly known as the Donders’constraint(Listing’s constraint in the case of eye movement).In this paper we show that in a neighborhood of the space of allowed orientations constrained by Donders’constraint,one can “l(fā)ocally linearize”the “rotation dynamics”by a static state feedback.The linear system,we show,has two components.The first one is transverse to the constraint-surface and the second one is tangential.Asymptotic stability of the transverse component guarantees(Donders’)constraint satisfaction.Controllers are synthesized on the linear system and mapped back onto the nonlinear head movement dynamics.

Let us now begin by addressing a long standing question,originally due to Donders:“To what extent is it true that head orientation is uniquely prescribed by the head pointing direction?”Simply speaking,head and eye rotations are constrained to have only two degrees of freedom and the conjecture is that their orientations are completely prescribed by their gaze or the frontal pointing directions.The constraint is imposed by requiring that,away from a suitably defined primary position,the axes of head/eye rotations(suitably scaled)are restricted to a fixed surface.For human head,this surface is called the Donders’Surface,see Medendorp et.al.[4]and Tweed[5].The Donders’surface(see[6]for a picture and a precise mathematical definition1Donders’surface constraint restricts the degree of freedom for head movement from 3 to 2.For eye movement,the corresponding Listing’s constraint,would restrict rotation of human eye about the direction of gaze.)is considered to be a fixed surface but changes from one human being to another.For human eye,not studied in this paper,Donders’surface degenerates to a plane.

In our earlier papers[7,8],we had introduced a Riemannian metric[9]on SO(3)and wrote down the associated Euler-Lagrange equation that describes the head rotation dynamics as a control system under the influence of an external torque.The Donders’constraint is imposed by constraining the state space to a submanifold DOND of SO(3).Alternatively,the problem of restricting the axes of rotation onto the Donders’surface is formulated as an optimal control problem[7]on SO(3),and the Donders’surface constraint is imposed as a penalty term.

Formulating and solving a control problem on a submanifold of SO(3),such as LIST[10]for the case of eye movement,provides a unique eye-orientation.This is because,under Listing’s constraint,eye orientation is uniquely specified by the eye-gaze direction2Eye orientation is uniquely specified at all but the backward gaze..Unlike what was perhaps conjectured by Donders,the same is not true for the head orientation under Donders’constraint.It has been shown earlier[10],[3]that under Donders’constraint alone,head orientation is not uniquely specified by the head pointing direction.In this paper we address this uniqueness question by imposing additional restrictions on SO(3).

The organization of the paper is now described.Section 2 introduces definitions and notations that are used later in the paper.In Section 3,we sketch and prove a version of the Donders’Uniqueness Theorem,when head rotations are restricted by a class of quadratic Donders’surface and the head pointing vectors are bounded by a maximal angle away from the straight pointing direction.In Section4,a rigid body dynamics is introduced that describes the rotation of the head.We show that this dynamics is locally feedback linearizable to a linear control system,for all points on the subset of the state space where Donders’Uniqueness Theorem was valid.The linear control system is now used to describe a Tracking and Path Following controller in Section 5.The path following control is next simulated in Section 6.In this section,we also sketch the Head-Gaits by graphing the movement of the Head-Top vector as the pointing vecor rotates in a circle.The paper concludes with a summary in Section 7.

2 SO(3)and quaternion

Throughout this paper,R denotes the set of real numbers and Rndenotes n dimensional real vectors.Snis the n dimensional unit sphere in Rn+1and Sn+denotes the upper hemisphere of Snwhose first element is positive.Next,we introduce the space SO(3)of orthogonal matrices and its connection to unit quaternion denoted by q.

2.1 Head orientation as a subset of SO(3)

Let R be a point in SO(3)with column vectors denoted by Ri,i=1,2,3.Columns of R can be associated with a frame attached to the head and would therefore cor-respond to a specific “head orientation”.Let us denote by{rij}i,j=1,2,3Previous papers of the second author[3](and references therein),had R3as the head-pointing direction.the components of the rotation matrix R.R1would be called the “head-pointing direction”,which is also the pointing direction of the nose.R3would be called the “head top direction”,which indicates if the head is tilting towards the left or right.We consider the head orientation in the right-handed system,denoted by symbol X-Y-Z,in R3(see Fig.1)3Previous papers of the second author[3](and references therein),had R3as the head-pointing direction..

Fig.1 The three columns of the rotation matrix R are shown attached to the head in R3.The vector R1is the pointing direction of the nose.The vector R3is the head-top direction.

We are interested in the following subset of SO(3)containing matrices,whose diagonal elements are all strictly positive

The restriction rii＞0 guarantees that the three vectors of the frame do not change hemisphere,as the head rotates.The imposed limitation on the head orientation,is quite reasonable and is dictated by the mechanical constraints of the neck muscles4The head-pointing vector cannot point backwards and the head-top vector cannot point downwards,etc..In this paper,we confine ourselves to a larger subset of SO(3)given by

where the set in(1)is a proper subset of H.

2.2 Some formula on H and quaternion connection

It turns out that the axis of rotation,for the rotation matrix R,can be written as

The variable α is the counterclockwise rotation angle about the rotational axisn,where cos(trace(R)?1),α ∈ [0,2π].Associated with a rotation matrix R,there is a unit quaternion q=[q0q1q2q3]T∈S3.This quaternion connection is well known in the literature(see[11]).A quaternion q and its antipodal point?q is mapped to a unique rotation matrix

The trace of the above matrix(4)is given by

The inversion formula to convert R(±q)∈ SO(3)to the corresponding pair of antipodal quaternions(see Section6.5 in[12])is given by

where n is in(3b)5Note that the inversion formula(6)is not valid outside the subset H of SO(3).When trace(R)≤0,there are several other ways to compute the unit quaternion q,see[11–13].They depend on the maximum component of the diagonal entries of the rotation matrix.,and where

2.3 Parametrization of unit quaternion

We now introduce a parametrization of unit quaternion proposed by Novelia and O’Reilly[14,15],given by

where ψ ∈ [0,π]and φ1,φ2∈ [?π,π).Compared to the well known Euler angle parametrization of unit quaternion[11],the Novelia-O’Reilly parametrization has certain advantages for the analysis we propose,specifically in representing the pointing direction of the head.We write down how the pointing direction R1is described under the parametrization(8).Substituting q(ψ,φ1,φ2)into the first column of(4)yields the expression

where we define a new angle parameter θ as

It is easy to see that r11=cosψ＞0 if 0≤ψ＜π/2.It follows that,for a fixed value of ψ,=sin2ψ,i.e,the pointing directions(r21,r31)are points on the circle with radius l=sinψ≥0 with the angle measure θ.

Remark 1According to the definition of θ there are infinitely many pairs of values of(φ1,φ2)satisfying θ= φ1+φ2and they yield orientations with identical pointing direction R1(ψ,θ)in(9).Each pair contributes to a specific choice of torsion for the orientation matrix,by ascertaining a specific position of R2and R3.

3 Quadratic Donders’constraint

Donders’constraint in a quadratic polynomial form was originally introduced in[16],[17]and describes a quadratic surface using quaternion parameters q0,q1,q2,q3,given by

where ciand cijare real numbers.Quadratic Donders’surfaces were subsequently used by the second author in[3]and[6].In this paper,we consider a simplified form(obtained by dropping the q0dependent terms from the right hand side of(11))of the quadratic Donders’surface given by

where we define the polynomial hDas

Remark2The simplified Donders’surface(12)contains Listing’s plane and the surfaces generated from Fick’s and Helmholtz’s gimbals,as a special case(summarized in Section6).Our motivation to consider this simplification is that the relative degree computations in Section4 is well defined entirely over a suitably chosen open set N.

We proceed to state and prove a theorem which shows that for a sufficiently small choice of the angle variable ψ,there exists orientations in H that satisfies(12).Rewriting(12)in terms of the parametrization(8)we obtain

We continue to assume that ψ is restricted to the in-in order to ensure that r11=cosψ ＞ 0.We also have a bound＜C,for all φ2in the closed interval[?π,π]where the constant C is assumed to satisfy

For the purpose of stating the next Theorem,we now define a constantas follows:

We now state and prove the following lemma,which is a precursor to the main theorem of this subsection.

Lemma 1Assume that the angle variables ψ and φ2are restricted to(ψ,φ2)∈[0,)×[?π,π),then there exist a unique value φ?1of φ1such that q(ψ,φ?1,φ2) ∈ S3+satisfies(12)withtrace(R(q(ψ,φ?1,φ2)))＞ 0 and the corresponding r11＞0.

ProofPlugging in the parametrization(8)into the Donders’constraint(12),we can easily solve for φ1and obtain

Note that the existence of φ?1follows easily from(14).We now proceed to show the trace condition.Observe from(15)thatwhich implies thatthe half angle rule andwe write

The trace condition follows from(5)and we obtain

Finally,from the above construction,it is easy to see that q(ψ,φ?1,φ2) ∈ S3+,since ψ is restricted to the interval[0,).We conclude that r11=cosψ＞0.This completes the proof.

Remark 3Geometrically,for a fixed value of ψ,there is a circle of possible pointing directions(see Fig.2).Each direction on a specific circle is completely specified by the angle variable θ introduced in(10).The following theorem claims that,as long as ψ is in the allowed interval[0,),every pointing direction is achievable by some orientation matrix in H that satisfies the Donders’constraint(12),and that this choice is unique.

Fig.2 The shaded region shows allowed region of pointing directions given by the inequality Points ofthe shaded region are on the surface of the unit sphere into the left of the plane X=cosP is the primary gaze direction and C is the current gaze direction.

Theorem 1(Donders’Uniqueness theorem) Assume that c22,c23and c33be specified fixed parameters of hD(q)in(13)describing the corresponding Donders’constraint.Let us defineas in(14)and choose ψ∈[0,).Then for a given pointing direction in the form p=R1(ψ,θ)from(9)where θ ∈ [?π,π),there exist a unique orientation matrix R∈H in the form R=[p,R2,R3]that satisfies the Donders’constraint(12).

ProofFirst of all,we can easily see that for 0≤ψ＜and?π ≤ θ ＜ π,the mapping(ψ,θ)R1(ψ,θ)∈S2+is bijective.Then,in order to prove this theorem,we need only to show the bijectivity of the mapping F:(ψ,φ2)(ψ,θ),for 0≤ ψ ＜and?π ≤ φ2＜ π,where θ is defined by

We show that for arbitrary θ,one can solve(16)for φ2.This is equivalent to asking if one can solve

for a suitable anglein the intervaland where the function φ?1is described in(15).We rewrite equation(17)as

For a given value of ψ and θ,and for φ1taking values in the intervalthe right hand side of(18)is a continuous function of φ1taking values in a closed interval[?1+ μ,1 ? μ](where μ is a small positive real number)and the left hand side of(18)takes all values in the interval(?1,1),it would follow that there is at least one value of φ1that would satisfy(18).Actual plotting of the left and the right hand side of(18)shows that this value of φ1is unique6It is easy to argue from the plots of the graph of functions in the l.h.s.and r.h.s.of(18)that the number of solutions of(18)is odd.On the other hand,it is known[3]that for a quadratic Donders’constraint,the number of orientations that match a specific pointing direction cannot exceed 2..The proof of this theorem now essentially follows from Lemma 1.

Remark 4For a fixed value of ψ the map F is a local diffeomorphism.To show this fact,it is enough to show thatis continuous and＞0 on its range because determinant of the Jacobian of F is given byBy differentiating(16)we obtain

Using the estimate

we now derive the following,using Lemma 1 from the appendix

4 A dynamical system for head rotation

We start this section by describing a subset Hof SO(3).Letbe as described in(14),we define

Intuitively,Hψcontains all orientation matrices in H whose pointing direction vectors are in the allowed region prescribed by Theorem 1.We now describe a rotating rigid body dynamics on Hψand show that such a dynamics is feedback linearizable by a static state feedback in the neighborhood of the Donders’surface.The rigid body dynamics and the output vector are described in the following subsection.

4.1 A rigid body dynamics with Donders’constraint

Let us consider the following rigid body dynamics[18,19]of the human head in the “inertial frame”

where J=diag{J1,J2,J3}is the moment of inertia matrix,Ji＞0,i=1,2,3;u∈R3are externally applied control torques;and ω =[ω1,ω2,ω3]T∈ R3is the angular velocity vector7The underlying state space of the dynamical system(20),(21)is Ξ=H×R3,a 6 dimensional manifold..We define

The Donders’constraint(12)can be written as

where n1,n2,n3have been defined in(3b),T in(7).To see(22),note that from(6)we have

Donders’constraint(22)easily follows from(12).An output function

is now introduced.

Remark 5As noted in the introductory Section1,the first two coordinates of the output function(23)uniquely specifies the head pointing vector R1in(9).The third coordinate of the output function is a scaler that measures the error derived from the Donders’constraint(22).This choice of the output function is dictated from the fact that,once the Multi Input Multi Output system is feedback linearized(sketched subsequently in Theorem 2),one can easily construct control signals that would drive the third coordinate of the output asy mptotically to zero(Donders’constraint satisfaction).Independently,the first two coordinates of the output function can be controlled to asymptotically track a prescribed signal that would make the head pointing direction,track a moving point target in R3.

Let us now define a proper subset HDas follows:

The subset HDcontains all orientation matrices in Hthat satisfies Donders’constraint(22).For notational convenience,we define ΞD=HD×R3,and define N to be a small enough open neighborhood of ΞDinside Ξ.

The dynamical system(20),(21),(23)can be written as a control-affine nonlinear 3-input 3-output system defined on N as follows:

where f and g are smooth vector fields N→R6,and all hiare smooth scalar functions N→R.Note that x abstractly denotes the state variable(ψ,θ,φ2,ω1,ω2,ω3)of the dynamical system.

4.2 Preliminary remarks from nonlinear MIMO control

We start with a standard definition of a decoupling matrix.

Definition 1(Vector relative degree[20]) The MIMO system defined by(25a),(25b)has a vector relative degree[r1,r2,r3]at a point x°∈ N if there exists a neighborhood U of x°and there exists a set of positive integers r1,r2,r3such that,for all x∈U,and for all i,j

and the decoupling matrix

has full rank.

We shall show that the dynamical system(20),(21)and(23)has a vector relative degree[2,2,2]at every point on N and the decoupling matrix(26)is a 3×3 matrix of rank 3 everywhere on N.

4.3 Main theorem(local feedback linearization)

Theorem 2On a small neighborhood of every point on N,there exist a static feedback law u?:=a(x)+b(x)v such that the rigid body dynamics(20),(21)with output(23)is feedback equivalent to the following controllable linear systems

where v∈R3is the new control input vector and new output variables are defined by

Remark 6In a small enough neighborhood of every point on N,Theorem 2 claims that up to a change of state variable and a static state feedback,the rigid body dynamical system can be reduced to a linear decoupled system with new states that are coordinates of the “pointing vector”and a scalar that measures how far away one is from the Donders’surface.

Remark 7For head to track a suitable trajectory of pointing directions,while satisfying the Donders’constraint,it would suffice to regulate the variable ξ to zero and use the two coordinates of(27)to implement tracking and path following controllers.

4.4 Proof of Theorem 2

ProofAccording to Theorem 1 in[21](see[20]),we need to show that the output h in(23)has a vector relative degree defined uniformly on ΞD.In order to calculate the relative degree,we need to compute Lie derivatives of h up to second order[20].The details of the Lie derivative computations have been omitted.

It is easy to see that Lgihj=0 for i,j=1,2,3.The decoupling matrix N in(26)is given by

Using the identity r11=r22r33?r23r32,it follows that

We claim that Lg1Lfh3(R)|＜0 so that detN(x)≠0 on HD.Standard Lie derivative computations show that

We proceed to evaluate the value of Lg1Lfh3(R),only on HD.Using(4)and(8),we obtain

Sinces in2it would follow that

Similarly on HD,using(6)and(8),the equation(31)can be rewritten as

Hence we have

where the lastinequality follows from the inequality(19)in Remark 4.Thus we are able to claim that the output h has uniformly,the relative degree[2,2,2]on HD.It would follow that we have the following local diffeomorphism and feedback law

where z3=˙z1,z4=˙z2,and ξ2=˙ξ18Note that ξ1= ξ,as in(29).,and

which renders the rigid body dynamics in(20),(21)locally feedback equivalent to the linear system in(27),(28),on ΞD.

Remark 8Local feedback linearization of rigid body dynamics described by a quaternion parametrized by Euler angles have been extensively studied in[22,23](see also[20]).It is equivalent to a direct inversion control,which renders dynamics of each Euler angle component a non-interacting linear system.In this paper we do not use the so called“inversion control of Euler angles’primarily for two reasons.The first one is the fact that Donders’constraint would still have to be written on the Euler angles and implemented.Secondly,it is a priori unclear if the relative degree constraint of[2,2,2]will be satisfied for all points in the set HD,for Euler angles as the new set of outputs.

Remark 9The linearization method adopted in this paper has been proposed in[21,24]in order to decompose the system dynamics into a tangential part and a part transverse to the Donders’submanifold(12).

5 Tracking and path-following control

In this section we consider two control laws,from linear control theory,called the reference signal tracking and the path following in order to control the head pointing direction(r21,r31).Our main purpose is to observe the head orientation as the head pointing vector is controlled.

5.1 Reference signal tracking

Already introduced in[20],we construct a control signal v(t)in(27)in order for zi(t)to track the reference signal Λi(t),i=1,2,and for ξ(t)to approach 0(see(29)).The control signal v(t)is constructed as follows:

for twice differentiable functions Λifor i=1,2 and the control gains(ci,di)for i=1,2,3 are chosen such that the origin of the linear system(27),(28),(34)is asymptotically stable.

5.2 Path following

In the“path-following"control design(see[21,24]),the state trajectory is made to approach and traverse along a predefined geometric path.We propose to design a controller for rest-to-rest maneuver control on HψD.Let χyand χzbe any two twice differentiable functions satisfying r221+χ2z(r21)＜sin2and χ2y(r31)+r231＜sin2respectively,i.e.,the points in(r21,χz(r21))and(χy(r31),r31)stay in an open circle of radius sin.

The path following controller is designed by replacing the output h1and h2in(29)with either of the following

and this replacement does not change the relative degree computed in Section4.The following proposition is a restatement of Theorem 2 for the new output(35)or(36).

Proposition 1On N there exist a static feedback law u?:=a(x)+b(x)v such that the rigid body dynamics(20),(21)with output(23),modified by the new h1and h2from(35)or(36),is locally feedback equivalent to the following controllable linear systems

where v∈R3is the new control input vector and new state variables are defined by either of the following

ProofThe proof of this proposition is omitted.It is easy to show that the determinant of a modified decoupling matrix is identical to that of(30).The vector relative degree structure does not change.

6 Simulation

The goal of this section is to show,by simulation,the changing orientation of the head as it is controlled for path following(Example 1),and as the head-pointing vector rotates in a circle(described in Example 2).Head orientations are visualized by displaying the three columns of the orientation matrix R(q)9The first column is the pointing vector and the last column is the head-top vector.on S2.

Example 1(Path following control using Fick-Gimbal)In this example(see Fig.3),we specialize the Donders’constraint to that prescribed by Fick-Gimbals10Fick Gimbals[10]were originally introduced by Fick as a simple model of head movement.The assumption here is that the head moves under pan and tilt.by setting c22=c33=0 and c23=?1 in(13).It would follow from(4)that r32=2(q0q1+q2q3)=0,i.e,the head side direction R2is restricted to the X-Y plane in R3.We assume that the moment of inertia matrix J is given by diag{0.2718,0.0529,0.2698}∈ R3×3and the control gain parameters in(34)are chosen as c1=c2=c3=100 and d1=d2=d3=20.We simulate path following and consider a path defined by χz(r21)=0.5sin(3πr21)in(35)with an initial orientation R(0)=I∈SO(3).The head orientation is assumed to start at rest,i.e.,ω(0)=0∈R3and the final value of the angular velocity is also observed to be at rest(see Fig.3(c)).In Fig.3(b),the three columns of the orientation matrix R are shown in three colors.Blue is the pointing direction(X axis),Green is the head top direction(Z axis)and Red is the head side(left ear)direction(Y axis).We observe that the movement of the head side direction is restricted to a plane,indicating that the Fick-Gimbal constraint has been imposed.

Fig.3 Path following trajectories of the orientation vectors R1,R2and R3and the angular velocity vector from Example 1 under Fick-Gimbal constraint.Observe that the side vector is restricted to the X-Y plane.(a)Pointing direction moving sinusoidally.(b)The three columns of the orientation matrix R are shown in three colors.(c)Head angular velocities under path following control.

Example 2(Head gait visualization) In this simulation,the orientations of the head is visualized by assuming that the pointing-vector rotates in a circle and we plot the head-top vector as the head points around the circle.Various choices of parameters that describes the Donders’surface are considered and the trajectories of the corresponding head-top vector plotted.

The head-pointing vector is assumed to follow the trajectory(r21,r31)=(0.5cosθ,0.5sinθ),as has been sketched in Fig.4(a).The following 5 cases are now considered.

Case I(Listing) In this case,we specialize the Donders’constraint to that prescribed by Listing11Listing’s constraint[10]was originally introduced by Listing as a model of eye movement,when head is assumed fixed.bysettingc22=c23=c33=0 in(13).We focus on behaviors of the head-topvector(r13,r23)which has been displayed in Fig.4(b),when the pointing-vector(r21,r31)goes around a circle as shown in Fig.4(a).The head-top vector follows the shape of afigure-of-eight.

Case II(Top and bottom part of the figure-of-eight tilted right or left) In this case,the parameters of the Donders’constraint(13)is chosen as c22=1,and c23=c33=0.The head-top vector maintains the shape of eight,but the top and the bottom part is shifted to the right as shown in Fig.4(c).It can be shown that when c22=?1,the head-top vector tilts likewise towards the left as the head pointing vector moves in a circle.

Case III(Center part of the figure-of-eight shifted right or left) In this case,the parameters of the Donders’constraint(13)is chosen as c33=1,and c23=c22=0.The head-top vector maintains the shape of eight,but its center is shifted to the right,as shown in Fig.4(d).It can be shown that when c33=?1,the head-top vector,analogously,maintains the figure-of-eight with its center shifted to the left.

Case IV(Fick-Gimbal and perturbed Fick-Gimbal)In this case,the parameters of the Donders’constraint(13)is chosen as c23= ?1.0(Fick-Gimbal)and c23= ?1.5(Perturbed Fick-Gimbal),while c22=c33=0.The headtop vector,sketched in Fig.4(e),rotates around the shape of eight and the direction of rotation is identical to the case of Listing discussed in Case I.

Case V(Helmholtz-Gimbal and perturbed Helmholtz-Gimbal) In this case,the parameters of the Donders’constraint(13)ischosenasc23=1.0andc23=1.5,while c22=c33=0.The head-top vector,sketched in Fig.4(f)for c23=1.5,rotates around the shape of eight and the direction of rotation is opposite to the case of Listing discussed in Case I.In the case of Helmholtz-Gimbal,when c23=1,(not sketched in Fig 4),the figure-of-eight degenerates to a line along the X axis12At this parameter the direction of rotation of the head-top vector reverses..

Fig.4 Trajectories of the head-top vector on the X-Y plane under various special cases of the Donders’constraint.(a)Trajectory of the head pointing-vector in Example 2.(b)Trajectory of the head-top vector under Listing’s constraint in Example 2,Case I.(c)Head-top vector trajectory from Example 2,Case II.Top and the bottom part of the figure of eight is shifted to the right.(d)Head-top vector trajectory from Example 2,Case III.Center of the figure of eight is shifted to the right.(e)Head-top vector trajectory from Example 2,Case IV,when c23is assumed to be?1.0 and ?1.5.This trajectory is qualitatively similar to the Listing’s case.(f)Head-top vector trajectory from Example 2,Case V,when c23is assumed to be 1.5.The figure of eight is oriented in opposite direction compared to the Listing’s case.

7Summary

We start this paper by revisiting Donders’question:“When is it true that the head orientation is uniquely determined by the head pointing direction?”A folklore conjecture,due to Donders,is that it is enough to impose Donders’law which restricts the rotation vectors of all admissible rotation matrices to lie on a surface.This conjecture is reasonable because it is true,for all but one eye-pointing direction,for the eye movement problem where the Donders’surface is replaced by a Listing’s plane.In this paper we restrict the set of admissible orientations to the subset H of SO(3)and the set of admissible pointing directions to a neighborhood of the straight gazing primary pointing direction,sketched in Fig.2.Under these restrictions,we show that Donders’conjecture is indeed true,when the Donders’surface is described by a simplified quadratic equation(13).

For the restrictive class of rotation matrices HDdescribed in(24),an important result in this paper is to show that in a small enough neighborhood N of ΞD,one can locally define a feedback equivalent linear system of the form(27)and(28).The states of the linear system are obtained using a local diffeomorphism Φ described in(32).The linear system can subsequently be used to synthesize tracking and path-following controllers,while the Donders’constraint is satisfied asymptotically.In Fig.3,a path following trajectory has been sketched.

Finally in Fig.4,we have visualized the movement of the head-top vector,as the front-pointing vector tracks a circular path.For various sub-cases of the Donders’surface,such a depiction of the head-gait is new.

Acknowledgements

The second author would like to acknowledge the contribution of Dr.Stefan Glasauer and Dr.Indika Wijayasinghe for their collaboration on the head movement problem,especially understanding of the Donders’constraint.

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13 March 2017;revised 9 April 2017;accepted 10 April 2017

DOIhttps://doi.org/10.1007/s11768-017-7034-9

?Corresponding author.

E-mail:bijoy.ghosh@ttu.edu.Tel.:+(314)4960586;fax:+(806)7421112.

This paper is dedicated to Professor T.J.Tarn on the occasion of his 80th birthday.

This work was supported by the Dick and Martha Brooks Professorship to Texas Tech University.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag GmbH Germany

Appendix

Lemma a1For all φ2∈ [?π,π),we have the following inequality

ProofSimple trigonometrical computations show that

Takafumi OKIwas born in Japan in 1980.He received his B.Sc.and M.Sc.degrees from Tokyo Denki University,Japan in 2004 and 2006 respectively,and his Ph.D.degree in Mathematics from Texas Tech University,Lubbock,Texas,U.S.A.,in 2016.His primary research interests are in nonlinear feedback stabilization and tracking,linear and nonlinear optimization problems.E-mail:takafumi.oki@ttu.edu.

Bijoy K.GHOSHreceived his Ph.D.degree in Engineering Sciences from the Decision and Control Group of the Division of Applied Sciences,Harvard University,Cambridge,MA,in 1983.From 1983 to 2007 Bijoy was with the Department of Electrical and Systems Engineering,Washington University,St.Louis,MO,U.S.A.,where he was a Professor and Director of the Center for BioCybernetics and Intelligent Systems.Currently,he is the Dick and Martha Brooks Regents Professor of Mathematics and Statistics at Texas Tech University,Lubbock,TX,U.S.A.He received the D.P.Eckmann award in 1988 from the American Automatic Control Council,the Japan Society for the Promotion of Sciences Invitation Fellowship in 1997,the Chinese Academy of Sciences Invitation Fellowship in 2016,the Indian Institute of Technology,Kharagpur,Distinguished Visiting Professorship in 2016.He became a Fellow of the IEEE in 2000,a Fellow of the International Federation on Automatic Control in 2014 and a Fellow of South Asia Institute of Science and Engineering in 2016.Bijoy had held visiting positions at Tokyo Institute of Technology,Osaka University and Tokyo Denki University,Japan,University of Padova in Italy,Royal Institute of Technology and Institut Mittag-Leffler,Stockholm,Sweden,Yale University,U.S.A.,Technical University of Munich,Germany,Chinese Academy of Sciences,China and Indian Institute of Technology,Kharagpur,India.Bijoy’s current research interest is in BioMechanics,Cyberphysical Systems and Control Problems in Rehabilitation Engineering.E-mail:bijoy.ghosh@ttu.edu.