Effects of Impedance Reduction of a Robot for Wrist Rehabilitation on Human Motor Strategies in Healthy Subjects during Pointing Tasks

Studies on human motor control demonstrated the existence of simplifying strategies (namely 'Donders' law') adopted to deal with kinematically redundant motor tasks. In recent research we showed that Donders' law also holds for the human wrist during pointing tasks and that it is heavily perturbed when interacting with a highly back-drivable state-of-the-art rehabilitation robot. We hypothesized that this depends on the excessive mechanical impedance of the pronation/supination (PS) joint of the robot and in this work we analyzed the effects of its reduction. To this end we deployed a basic force control scheme, which minimizes human–robot interaction force. This resulted in a 70% reduction of the inertia in the PS joint, and in decrease of 81 and 78% of the interaction torques during 1- and 3-d.o.f. tasks. To assess the effects on human motor strategies, pointing tasks were performed by three subjects with a lightweight handheld device, interacting with the robot using its standard PD control (setting impedance to zero) and with the force-controlled robot. We quantified Donders' law as two-dimensional surfaces in the three-dimensional configuration space of rotations. Results revealed that the subject-specific features of Donders' surfaces reappeared after the reduction of robot impedance obtained via the force control.

During traditional neurorehabilitation sessions the role of the patient is quite passive and the participation of the high-level structures of his/her nervous system are limited. On the other hand, robotic therapy allows subjects to perform active movements, thus promoting neurogenesis and neuroplasticity, and optimizing functional recovery after neurological injuries [8,9].
During therapy rehabilitation robots are meant to: (i) provide assistance as needed, not to perturb voluntary movements, and to help completing tasks that patients are not able to perform autonomously, and (ii) apply, if required, resistive force fields to improve the efficacy of the treatment. During assessment robots have to be as transparent as possible in order to minimally interfere with human natural motor strategies. In this sense a high back-drivability, i.e., a low perceived mechanical impedance during user-driven motion, is of crucial importance. Therapy and assessment sessions can be interlaced, and assessment sessions may determine the amount of future therapy to be imparted; therefore, assessment may also have an indirect impact on therapy. The technological choices regarding the design of mechanical structure and control strategies for this kind of robots have to face several issues, such as safety, dynamical adaptability and biomechanical compatibility [10]. Nevertheless, besides biomechanical constraints, also the understanding of neural strategies behind human motor control (especially in the presence of dynamic interactions) should be considered to define some indications for robots design.
In this regard, it has been shown that rehabilitation machines have to comply not only with 'hard' (biomechanical) constraints but also with 'soft' (neural) constraints [11], which are adopted by the brain to solve kinematic redundancy and to implement strategies for motor efficiency.
Studies on human motor control have demonstrated the existence of simplifying strategies adopted by the brain when dealing with kinematically redundant problems. Such strategies are often referred to as Donders' law or Listing's law for historical reasons, in relation to the early Donders' and Listing's studies on the eye movements. In a good approximation, the eye can be considered as a center-fixed sphere rotated by the action of six (i.e., three agonist-antagonist couples) extraocular muscles (EOMs). EOMs provide 3-d.o.f. kinematics allowing full mobility in the space of rigid body rotations with the only limitations given in terms of the range of motion. When looking at some point in space, the gaze direction is fully determined but not the amount of ocular torsion about the line of sight. In other words, for a given line of sight, uncountably many eye configurations (torsions) exist that correspond to the same gaze direction.
In 1847, Donders experimentally found that for a given steady gaze direction there is only one eye configuration (Donders' law) [12]. In other words, physiological eye configurations are described by a two-dimensional (2-D) surface embedded into the three-dimensional (3-D) space of eye configurations: a solution to redun-dancy. Two decades later, Listing and Helmholtz determinated that such a 2-D surface is actually a plane: the eye assumes only those positions that can be reached from the primary position by a single rotation about an axis in the Listing's plane, which lies orthogonal to the gaze direction in primary position (Listing's law) [12]. Listing's law is therefore a particular form of a more general Donders' law: the former prescribes a plane, whereas the latter prescribes a generic surface. Hereafter we shall refer, in general, to Donders' law.
It had been debated for quite some time whether Donders' law was due to a biomechanical or a neural mechanism. It is now known that such a law holds during fixation, saccades, smooth pursuit and vergence, but not during sleep and vestibuloocular reflex, suggesting that it is actively implemented by a neural mechanism [13]. Moreover, in the last two decades, Donders' law has been found to apply to the head as well as to limb movements [14,15].
Recent studies demonstrated that Donders' law applies also to wrist movements during pointing tasks (in which the 3 d.o.f. of the wrist are redundant) [16]. This study was conducted measuring wrist orientations during pointing movements using a lightweight handheld device, which introduces a negligible perturbation to the physiological movements. The 3-D rotations vectors that resulted were well fitted by quadratic (Donders') surfaces, whose curvature expresses inter-subjects differences in motor strategies, thus denoting a personal 'style' in solving redundancy [16].
The same analysis was conducted in Ref. [11] with subjects performing similar pointing tasks while interacting with the InMotion 3 system (Interactive Motion Technologies) -a state-of-the-art robot for wrist neurorehabilitation, specifically designed to comply with the biomechanical constraints and providing a high level of back-drivability [17]. In this case no statistically significant difference was found between subjects, suggesting that the robot flattens out any personal motor strategy; in particular, it was hypothesized that the impedance of the most proximal joint of the robot, responsible for pronation/supination (PS) movements, was not low enough to comply with the neural constraints. In other terms, the robot perturbs voluntary movements of the subjects, despite its high level of transparency.
The objective of this study is to analyze the effect of a reduced perceived impedance of the InMotion 3 system on human motor strategies during pointing tasks. Our work could allow us to examine the effects of different levels of back-drivability on human natural movements and to point out which level can avoid the occurrence of constraining behaviors, thus suggesting a new approach for the design of rehabilitation robots that, at the state of the art, do not consider neural constraints but only biomechanical ones.
To this aim the handle of the robot was provided with a load cell and a basic force control scheme (with a null reference) was implemented to minimize the interaction force exerted by the user. This kind of approach has been already validated in Ref. [18] on a different platform, the MIT-MANUS system [1], in both simulation and experimental tests. In Ref. [18] a reduction of (mean and maximum) forces exerted by a subject during planar pointing tasks has been verified in different operative conditions, demonstrating the effectiveness of the chosen force control scheme in improving the system transparency.
In this work inertia, friction and gravity effects were estimated for the PS joint of the InMotion 3 system to be actively compensated for. The performance of the robot has been analyzed with and without force control to verify the reduction of its mechanical impedance. Moreover, the reduction of forces/torques required by three healthy subjects to perform two different kinds of tasks (involving, respectively, 1or 3-d.o.f. movements) has also been measured.
The effect of the reduced impedance of the robot on human motor strategies during 3-d.o.f. pointing tasks has been analyzed, by assessing wrist rotations of three healthy subjects in three different testing conditions: movements with (i) the handheld device, (ii) the robot with its standard PD control with gains set to zero and (iii) the force-controlled robot; for each experiment, Donders' surfaces were calculated as performance index.
Section 2 provides a description of the InMotion 3 wrist robot, while the force control scheme is described in Section 3 together with the estimation of robot dynamic parameters and the calculation of kinematic operators used for its implementation. The effects of force control on robot transparency and on human wrist kinematics/neural motor strategies are presented in Sections 4 and 5, respectively.

InMotion 3 Wrist Robot
The InMotion 3 system [17] has 3 active d.o.f.: PS, abduction/adduction (AA) and flexion/extension (FE). A differential mechanism is used to achieve AA and FE movements; PS rotations entail the sliding along a curve guide of the housing comprising the differential gearing and the AA and FE actuators (Fig. 1). In Ref. [17], a quantification of the three axis inertia, friction and gravity terms is provided showing that PS values are one order of magnitude higher with respect to the other 2 d.o.f. The robot is actuated by three DC motors that are equipped with rotary encoders; angle resolution measurement is 0.017 • for the PS d.o.f. and 0.026 • for FE and AA d.o.f.
At the basis of the handle 2 passive d.o.f., a linear slider and a revolute joint, are used to comply with the anatomic characteristics of different subjects and also for a further adaptation of the robot to the axes of rotation of the human wrist (Fig. 1).
The robot is provided by the manufacturer with a PD + gravity compensation control scheme (which, for simplicity, will be referred as PD control) so as to generate programmable viscoelastic torque fields to deliver forces or impart movements during therapy. In particular, it can be programmed to provide different levels of mechanical impedance by properly setting the control stiffness and damping parameters (see Ref. [17] for details on the standard control of the InMotion 3 system). The minimal impedance, with this control scheme, can be achieved when these pa-  rameters are both set to zero; in this case the only role of the motors is to compensate for the gravity effect.
In this work we sensorized the robot handle, in order to implement a direct force control for a further reduction of the impedance of the robot PS joint. To this aim we decided to remove the linear slider and the passive revolute joint, whose movements could not be measured, and to collocate a load cell to the basis of the end-effector (Fig. 2).
A six-axis JR3 torque/force sensor (20E12A4-I25-EF 100N5, JR3; force f.s.: 100 N; torque f.s.: 5 Nm) with external electronics was used. The six analog outputs of the sensor were directly connected to an acquisition board already present on the robot, which automatically synchronizes external signals with the 200-Hz control software. The sampled signals have been multiplied by a calibration and decoupling matrix, provided by the manufacturer of the sensor.

Force Control Implementation
The control algorithm used in this work is adapted from Refs [19,20]. It compensates for the robot dynamics (gravity, inertia and friction) and minimizes the end-point interaction force since a zero reference is imposed.
The equation of robot dynamics (Coriolis and centrifugal effects have been neglected because of the low values of velocity involved in the wrist pointing tasks) can be expressed as: where F int represents the 6 × 1 vector of the generalized forces exerted by the user (with respect to the base frame), b is a 3 × 3 diagonal matrix of constant friction coefficients, B(q) is the 3 × 3 inertia matrix, J T (q) is the 3 × 6 transpose Jacobian matrix, g(q) is the 3 × 1 gravity term and τ is the 3 × 1 vector of control torques.
Relation (1) is composed by three scalar equations, which refer to PS, AA and FE joints, respectively; thus, the vector of joint coordinates is The implemented control law is: where the symbolˆindicates an estimation of the corresponding vector or matrix, F int indicates the 6 × 1 vector of generalized forces measured by the load cell and K P is a 3 × 3 positive-definite diagonal gain matrix. Force error can be indicated as where F d is the desired interaction force. By substituting (2) in (1) and assuming that robot dynamics are perfectly estimated, the force error dynamics becomes: Considering M = K −1 P , (3) can be expressed as: If F d is set to zero, (4) indicates that the system behaves like a free body with mass M, so that increasing proportional gain (within stability margins) will reduce the perceived inertia.
Force control in (2) has been implemented only for the PS joint since the AA and FE joints are already highly back-drivable (their mechanical impedance is one order of magnitude smaller than the one of PS d.o.f., [17]); therefore, the control torques for the 3 d.o.f. can be written as: with where τ g PS = g PS sin(q PS ) and b PS are the gravity term and the friction coefficient of the PS d.o.f., respectively.
Indicating for simplicity τ int The implementation of the control scheme in (7) required the calculation of the robot kinematic operators and the estimation of its dynamical properties.

Robot Kinematics
Since the robot native control is implemented only in the joint space, the calculation of the Jacobian matrix J(q), and of its transpose J T (q), was necessary to map the torques/forces measured in the operational space in the corresponding actions on the joints. A schematic representation of the robot kinematics is provided in Fig. 3. The Denavit-Hartenberg convention was used to select the frames of reference (the parameters are reported in Table 1) and to calculate the forward kinematics. The adjoint transformation [21] was used to map the vector of generalized forces from the the sensor coordinate frame (indicated with S in Fig. 3) to the global reference system (indicated with O in Fig. 3).
Let p os be the 3 × 1 position vector of the origin of frame S from the origin of frame O and R os the 3 × 3 rotation matrix describing the orientation of frame S with respect to frame O. This matrix is calculated as: being the matrix associated with the ith rotation of the ith frame with respect to the (i − 1)th one. Rotation matrices R o1 (t 1 ) and R 2s (t 3 ) indicate the orientation of frame 1 with respect to frame O and of frame S with respect to frame 2, respectively.
Indicating with g os the pair (p os , R os ) the matrix: Ad g os = R os p os ∧ R os 0 R os (9) associated with g os is the adjoint matrix; its transpose transforms force vectors from the frame S the frame O as follows: It has been assumed that the vector without superscripts refers to the base frame.

Gravity Term Estimation
To estimate the gravity term for the PS joint, a correlation between different static configurations and the torques required to hold the robot in those positions was performed.
It is important to note that tests were carried out with AA and FE joints locked, being the actuators commanded to maintain their 0 reference position through highgain PD position control (in these conditions we verified a deviation from the rest position of FE and AA of less than of 3 • , which should not sensibly affect the estimation). The choice of neglecting the contribution of changes in AA and FE configurations on the PS gravity term is also adopted in Ref. [17] since the load produced by the handle on the PS joint is reasonably independent of the AA and FE orientations of the robot in the range of operation.
The robot was commanded to reach 10 positions (from −0.6 to +0.6 rad with respect to its rest position); PS actuator torques exerted to maintain these rotations in static conditions were evaluated. Figure 4 shows an example of the PS torque and angle patterns during one estimation test of the gravity term.
As in Ref. [17] the gravity term has been assumed sinusoidally varying with PS angle so a linear regression between PS torque and the sine of the PS rotation was calculated for five trials. In Fig. 5, the collected data (static PS angles and torques for the five trials) are shown together with the best-fitting curve.  The slope of the linear curve in Fig. 5, averaged on the five trials, was found to be g PS = 2.254 ± 0.009 Nm, with a mean R 2 of 0.980.

Inertia and Friction Estimation
To estimate inertia and friction terms the robot was back-driven to perform a few rapid oscillating movements. The PS motor compensated for the gravity effect (on the basis of the estimation reported in Section 3.2.1), and the AA and FE actuators were commanded to maintain their rest positions through high-gain PD position control; in these conditions a deviation of less than 3 • for FE and AA was found. Force/torques exerted on the handle were recorded and their effect on the PS joint was calculated as: We estimated the inertia and friction terms for the PS joint by regressing the τ PS sens reported in (11) with angular velocities and accelerations of the PS d.o.f., as reported in: The multivariable linear regression reported in (12) was calculated on five trials; the best-fitting plane is shown in Fig. 6. We found τ PS 0 = 0.219 ± 0.049 Nm, b PS = 0.055 ± 0.005 Nm · s/rad and B PS = 0.012 ± 0.001 kg · m 2 with a mean R 2 of 0.942.

Effects of Force Control on Robot Transparency
In this section the effectiveness of the force control in (7) in improving the transparency of the robot is tested. First, the reduction of the robot mechanical impedance in the PS joint due to the action of the force control has been estimated; then the reductions of forces/torques exerted by three subjects while performing two kinds of pointing tasks were analyzed, comparing the cases of unpowered robot (UR), robot with zero-gain PD control (ZPD) and force-controlled robot (FC).

Reduction of PS Joint Impedance
To evaluate the perceived robot mechanical impedance with the action of the force control, the same procedure described in Section 3.2.2 was followed. In particular, Figure 6. Multivariable linear regression for PS inertia and friction estimation as reported in (12).
Considering the values averaged on five trials, we found τ PS 0 = 0.219 ± 0.049 Nm, a viscous friction b PS = 0.055 ± 0.005 Nm · s/rad and an inertia B PS = 0.012 ± 0.001 kg · m 2 , with a mean R 2 of 0.942. The multivariable linear regression reported in (12) was calculated on five trials; the best-fitting plane is shown in Fig. 7. We found τ PS 0 = −0.059 ± 0.043 Nm, b PS = −0.002 ± 0.002 Nm · s/rad and B PS = 0.0037 ± 0.0001 kg · m 2 with a mean R 2 of 0.980.
The negative value of friction suggests that an overestimation of this parameter was made. Anyway, this value can be considered negligible since it was found to be not statistically different from zero.
Of note, the implemented force control scheme in these conditions causes a reduction of the inertia of about 70% with respect to the ZPD case (simple gravity compensation, as described in Section 2). In particular, the estimated inertia B PS = 0.0037 kg · m 2 is consistent with the expected value of 0.004 kg · m 2 , demonstrating the effectiveness of the force control action.

Reduction of Interaction Forces/Torques
In this section PS torques (τ PS sens , as in (11) Figure 8 shows the PS torque (11) exerted by one representative subject in the different testing conditions. The only gravity compensation (PD control with gains set to zero) reduces the torques required to perform the task with respect to the case when motors are turned off; a further decrease of the torque in the case of the forcecontrolled robot can be noticed. Figure 9 reports the peak and mean values of the PS torque for the 1-d.o.f. trials on the three subjects.
It can be seen that these values slightly decrease, increasing the gain K P , although no major changes occur. In the case of K P = 300 kg −1 · m −2 the peak torque decreases to 90% while the mean torque of the 81% with respect to the UR case. PD control only causes a reduction of 54 and 46% of the peak and mean torques, respectively. Of note, the force control (K P = 300 kg −1 · m −2 ) reduces the peak and mean torques of 77 and 65%, respectively, with respect to the PD control condition.

The 3-d.o.f. Task
During this 3-d.o.f. task, three healthy subjects were asked to perform pointing movements with the wrist from the center of the workspace to eight peripheral  points arranged in a circle displayed on the monitor; this is also a typical 'video game' rehabilitation exercise (often referred to as clock game), which has been used in Ref. [11] to test the effect of the robot on human motor strategies. Figure 10 shows the PS torque (11) exerted by one representative subject in the different testing conditions. Also in this case gravity compensation PD control with gains set to zero lowers the PS torques with respect to the UR and FC case improves this effect. Figure 11 reports the peak and mean values of the PS torque for the trials on the three subjects. No major changes occur with the gain increasing. In the case of K P = 300 kg −1 m −2 the peak torque decreases to 80% and the mean torque decreases to 78% with respect to the UR case. PD control only decreases to 5.6 and 5.4% the peak and mean torques, respectively; thus the force control (K P = 300 kg −1 m −2 ) reduces the peak and mean torques 79 and 77%, respectively, with respect to the PD control condition.
An overall view of the results obtained on the three subjects is presented in Table 2.  To verify that the decrease of the interaction torques was really due to a reduction of the robot impedance and not to possible differences in the execution of the tasks during the five testing conditions, the RMS values of the velocity for the three subjects were analyzed. Results reported in Tables 3-5 show that the tasks executed by each subjects in different conditions are comparable and that the ones executed with lower velocities are actually those where the robot is in UR or ZPD conditions. These observations confirm that the reduction of the interaction torques in the FC condition were due to a real reduction of robot impedance.

Considerations on Force Control Performance
The performance of the adopted force control scheme in general depends on the dynamic characteristics of the robot and some unstable behaviors might occur when control gains are high. This is particularly true in the case of industrial robots, where the intrinsic mechanical impedance is extremely high. Nevertheless, in our specific case the robot already presents a good level of back-drivability, so that when we implemented and tested the force control action no critical stability problems occurred. We tested the robot with a maximum value of K P = 300 kg −1 · m −2 , corresponding to a very low perceived inertia (about 0.003 kg · m 2 ), and we did not experienced any unstable behaviors, not even when the handle was manually excited to oscillate with a peak velocity of the order of 10 rad/s and a peak acceleration of the order of 150 rad/s 2 (values that are grater than the values normally measured during pointing tasks). Nonetheless, trials for the evaluation of human motor strategies (which will be described in Section 5) were performed using a gain value lower than the maximum tested value.
In both 1-and 3-d.o.f. tasks we found a net decrease of torques exerted by the subjects in case of the force-controlled robot with respect to the PD control condition while no significant differences occurred when different gain values were used for force control. Since improving the gain values, with a fixed friction compensation, corresponds to reducing only the perceived inertia (0.007 and 0.003 kg · m 2 for K P = 150 and 300 kg −1 · m −2 , respectively) it could be concluded that the major contribution in the achievement of robot transparency is given by the friction reduction (which is of course due to the direct friction compensation, but also to the force feedback).
Moreover, considering a perfect estimation of gravity and friction terms, from (1) and (2) it can be seen that the force error dynamics becomes: It is worth recalling that B represents robot inertia,B is its estimation andq is the vector of joint accelerations. Equation (13) shows that an error in the inertia estimation is compensated for by the proportional gain. If inertia is overestimated the force control gain has to be reduced by the same amount to preserve the stability margin, while if inertia is underestimated the gain can be increased. From a different perspective we can say that the productBK P is responsible for the stability so that a variation of the gain makes the control law basically insensitive to inertia estimation.

Human Motor Strategies during Pointing Tasks
In this section we analyze the pointing tasks of three subjects in three different operative conditions. First, wrist rotations were recorded using a handheld device that introduced a negligible loading effect (in this condition subjects movements were basically free); these results were compared with the ones obtained during the interaction with the robot when the gains of its standard PD control scheme were set to zero and when the proposed force control was active. Since the zero-gain PD control actually corresponds to only gravity compensation, the comparison with the force-controlled robot allows us to also evaluate the effect of inertia and friction compensation.
For each condition the experimental protocol included five learning trials that were used to acquaint the subjects with the experimental setup and five trials to be used for data analysis.

Handheld Device
To assess natural motor strategies during pointing tasks with the wrist, we used a 'video game' showing subjects some visual targets to be reached (from/to a central target to/from peripheral ones) (Fig. 12); this protocol has been already used to assess the existence of Donders' law for the wrist during pointing tasks [16].
The subject under test was seated on a chair, and rested his/her arm on a support purposively created to bind the arm and forearm (Fig. 13). In this way she/he could execute pointing tasks moving only the wrist.
Each trial started after defining a primary position (see Ref. [16] for details); the subject was then instructed to move the round cursor on the screen towards the peripheral positions randomly chosen by the software among 1, 2, . . . , 8 (Fig. 12) and then back to the central position. To register the wrist movements we instrumented an hollow cylindrical handle (height: 150 mm; outer diameter: 50 mm; inner diameter: 35 mm; mass: 120 g) with an inertial magnetic unit (IMU, MTx-28A53G25 device from XSens; angular resolution: 0.05 • ; static orientation accuracy: below  1 • ; dynamic accuracy: 2 • RMS; bandwidth: 40 Hz); this setup is indicated as the handheld device (Fig. 13).
With the wrist in its primary position, a fixed reference frame {x 0 , y 0 , z 0 } was defined (see Fig. 13): z 0 -axis along the vertical direction (upwards), x 0 -axis horizontal and aligned with the forearm (forward), and y 0 -axis horizontal and perpendicular to the forearm (leftward). A second (moving) reference frame {x, y, z} attached to the wrist (see also Fig. 12) was defined to coincide with the fixed reference frame when the wrist is in the primary position.
The IMU, connected to a PC, was configured to continuously acquire a rotation matrix R (at a rate of 100 samples/s) that expresses the orientation of the wrist moving frame with respect to the fixed frame.
We calculated the rotation vector as: where R j,k indicates the (j, k) element of the matrix R and θ = arccos[(trace(R) − 1)/2], being trace(R) = R 1,1 + R 2,2 + R 3,3 and |θ | < π. The vector r defines the axis (parallel to r itself) and the amount of rotation ( r = θ ). Since the primary position was selected as the 'home' position for the IMU the coordinates of the x-, y-and z-axis with respect to the fixed frame could be determined, respectively, as the first, second and third columns of the matrix R. For this reason the pointing vector n (which represents the pointing direction during the tasks) was always parallel to the moving x-axis after the reset procedure and it could be determined as: While the three components of a rotation vector r, in general, define points of a 3-D space, the wrist pointing vectors n necessarily lie in a 2-D space.
Considering the subscript i to represent the ith acquisition sample, the temporal sequence of acquired rotation matrices and of the derived rotation and pointing vectors can be indicated as R i , r i and n i , respectively.
Numerically, the sequence r i = [r xi r yi r zi ] T was fitted to r * i = [r * xi r yi r zi ] T , where r * xi is defined by a generic quadratic surface: where the coefficients C 1 , . . . , C 6 were determined via least-squares fitting methods. The first three coefficients (C 1 , C 2 and C 3 ) define a plane, while the last three coefficients (C 4 , C 5 and C 6 ) are related to the curvature of the fitted surface. In particular, the coefficient C 5 denotes the amount of twisting of a quadratic surface. Figure 14 reports an example for the graphical representation of the temporal sequences of rotation and pointing vectors (r i and n i ). Remarkably, it can be observed that the rotation vectors tend to lie on a 2-D (Donders') surface (represented by (16)), which can also be well approximated by a plane near the primary position. As in previous studies [11,16], the thickness of a Donders' surface is defined as the standard deviation of the fitting error between the sequence r i and the bestfitting surface. For a given set of fitting coefficients (C 1 , . . . , C 6 ), the fitting error is defined as r xi − r * xi . Obviously, thickness values quantify the goodness of fit and indicate how much the rotation vectors tend to lie on the best-fitting surface (Donders' surface); low values of thickness indicate that a soft constraint (such as Donders' law) applies to wrist kinematics during pointing tasks. As an example, both sequences of a single trial are plotted in Fig. 14. While the wrist pointing directions necessarily lie in a 2-D space, the three components r xi , r yi and r zi of a rotation vector r i , in general, define points of a 3-D space. Remarkably, it can be observed that the rotation vectors tend to lie on a 2-D surface (Donders' law), which can be well approximated by a plane near the 'zero' position (Fig. 14).

Wrist Robot
Subjects were strapped to a chair with their right arm fasten to the InMotion 3 armsupport by appropriate belts to minimize torso, shoulder and elbow movements; in this configuration only wrist rotations were left unconstrained while the subject grasps the handle of the robot (Fig. 15). The more distal strap only constrains the flexion of the elbow while allowing a free PS movement. In this way the PS joint is moved by the user only exerting forces on the handle and not via the strap.
For the experiments described in this section, two different conditions were tested: in the first one, the motors of the robot were commanded so that they only compensate gravity, with the stiffness and damping parameters of the native impedance controller set to zero; in the second one, the robot was controlled using the force control law in (2). In what follows we refer to these two operative conditions as zero-gain PD control and force-controlled robot, respectively.
The starting position for the wrist was set to coincide with the mechanical zero position of the robot, defined in Ref. [17]. Knowledge of the sequence, from prox- imal to distal, of the mechanical joints allows us to determine the wrist orientation R w from the angles as sensed by the encoders of the system: where x, y and z coincide with the axes of the fixed reference frame, i.e., the primary or 'zero' position for the wrist. For a given wrist orientation R w , the wrist pointing direction n i is given as in (15). During the execution of each task, data were acquired at a rate of 200 samples/s, in particular the three sequences of sampled angles, and (where i refers to the ith sample at time t i ). Using (17), the sequence of wrist orientation R wi at time t i was derived. The sequence of wrist pointing directions n i was evaluated via (15) and finally the sequence of rotation vectors r i was obtained via (14). The sequence r i = [r xi r yi r zi ] T was numerically fitted to a generic quadratic surface with coefficients C 1 , . . . , C 6 as in (16).

Free Movements and Human-Robot Interaction: A Comparison
Donders' surfaces are typically subject-dependent and reveal personal motor strategies, as demonstrated in previous studies [16]. The effect of a purely mechanical (holonomic) constraint on a 3-d.o.f. wrist mechanism (e.g., a locked joint) is to constrain the system configuration exactly to a 2-D surface. Noisy measurements and mechanism non-idealities (e.g., backlash) would reveal such a surface with a certain resolution, for the robotic systems under consideration this is (often much) less than 0.1 • . Observing a mechanically unconstrained 3-d.o.f. system evolving on a subspace (2-D surface) of its configuration space (3-D) is indicative of active control. This has been attributed to a form of neural control in studies relative to the oculomotor system [12] and to the upper limbs [14][15][16]. Unlike physical constraints, neural control is characterized by much larger variability (or 'biological noise' [12]) leading to thicknesses of the order of 1-3 • [12,14].
In this section we reported the results obtained in the three different experimental conditions, namely experiments with the handheld device, with the robot in zero-gain PD control mode and with the force-controlled robot. In Figs 16 and 17 the Donders' surfaces and the histograms of the mean values of C 1 , . . . , C 6 coefficients are reported.
In the experiments when the handheld device was used, neural kinematic constraints arose as 2-D surfaces embedded in the 3-D space of wrist configurations. The lightness of the handheld device allows the subjects' own motor style to emerge: different trials show similar curvature for the same subject, while they differ among subjects, as already proved in Refs [11,16]. During the interaction of subjects with the robot in zero-gain PD control mode the high goodness-of-fit in terms of thickness (1-2 orders of magnitude smaller than in the experiments with the handheld device) revealed the strong influence of the robot in covering the human motor strategies. In this case 2-D surfaces are similar for all subjects, highlighting the absence of any biological noise (Fig. 16). It has to be noticed that, since the the robot constraining behavior has also been verified in Ref. [11] (where experiments were conducted without any modification to the mechanical structure of the system), the perturbation to the human natural motor strategies cannot be due to the removal of the passive revolute joint and of the linear slider.
In the case of the force-controlled robot, 2-D surfaces were different among tested subjects; moreover, each subject' surfaces were similar to those coming from the experiment with the handheld device, with thickness values back into the physiological range (1-3 • ). From the analysis of the coefficients C 4 , C 5 and C 6 , which are related to Donders' surfaces curvature, no discrepancy between the values measured with the handheld device and with the force controlled robot emerges. This confirms the analogies between the two performances.
The constraining action of the robot, when controlled through the PD control (with gains set to zero) is particularly evident in the Helmholtz-like behavior of the surfaces: the sign of the C 5 coefficient is inter-subjectively invariant and positive. It is worth recalling that the Helmholtz gimbal is a 2-D mechanism and its rotation vectors necessarily lie on a 2-D surface; in the case the surface is expressed through (16) this implies C 5 = 1/2 and all the other coefficients to be zero. The explanation for this result is that subjects' natural movements are affected by an excessive mechanical impedance on the PS joint of the robot; this is confirmed by the restoration of a natural behavior during the execution of the tasks with a more transparent robot (from Table 6 it can be seen that thickness values became physiological) and by an evident activation of the PS d.o.f.

Conclusions
The paper addresses not only the traditional issues of physical human-robot interaction in terms of safety and biomechanical compatibility, but also the problem of coping with human motor strategies and neural mechanisms adopted to solve redundancy during motor tasks that are not appropriately considered in the design of actual robotic systems.   Previous studies reported the existence of neural constraints for the human wrist during kinematically redundant tasks and also showed how such motor strategies can be perturbed when subjects interact with a state-of-the-art wrist robot, despite its high level of back-drivability [11,16].
It was hypothesized that the robot mechanical impedance associated with the PS joint could be directly responsible for the perturbation of the natural motor strategies. For this reason in this work we implemented a basic control scheme, specifically for the PS axis, to reduce the human-robot interaction forces and to study the effects of a lower perceived impedance on such neural constraints. From the analysis of Donders' surfaces associated with the execution of pointing tasks we found that, when interacting with the force-controlled robot, subjects regained nat-ural motor behaviors as during free pointing (measured with a lightweight handheld device).
The work emphasizes the possibility of minimizing the perturbation to human natural motion even through simple solutions like a well-known direct force control approach. This represents the starting point towards new scenarios for the design of rehabilitation robots, that at the state of art do not consider neural constraints but only biomechanical ones. In this regard, our future work will aim at investigating the range of impedance values necessary to fully cope with neural constraints; to the authors' knowledge this aspect has not been analyzed yet and it can represent an important criteria for robot design.