The Hbot : A Holonomic Spherical Haptic Interface Driven by Non-Holonomic Wheels

The Hbot : A Holonomic Spherical Haptic Interface Driven by Non-Holonomic Wheels

Abstract

We present the Hbot, a holonomic, singularity-free spherical robot designed for haptic simulations. The Hbot is made up of a caged sphere actuated by steered and driven non-holonomic wheels to produce continuous and unlimited spherical motions. We analyse the kinematic interface between a sphere and arbitrarily positioned, steered and driven non-holonomic wheels in the general case. We also present a detailed singularity analysis and show that workspace-boundary and workspace-interior singularities can both be avoided at the design stage. We implement a prototype using two steered and driven non-holonomic wheels and show experimental results for trajectory tracking and rendering of various rotational stiffness levels.

Figure 1: The relevant frames and points to define the kinematic interface between a sphere and arbitrary driven and steered non-holonomic wheels

I Introduction

Haptic devices are essential in our world today with applications ranging from surgery to search and rescue. They have become even more important for tasks where visual information alone is not sufficient and may result in undesired errors. A typical example is minimally invasive surgical training where haptic feedback has improved the training performance of surgeons [1]. Haptic devices can have different types of workspace requirements depending on the desired application. In this work, we are interested in the generation of continuous and unlimited spherical motions for haptic simulations.

A flight simulator [2] for example, requires a mechanism with unlimited spherical motions to achieve a meaningful performance. Moreover, in neuro-physiological experimental studies [3], a device with unlimited spherical motion is required to study the contribution of vision and proprioception to the perception and control of hand orientations in three-dimensional orientation-matching tasks. Currently, spherical motions are produced either by spherical motors or more general mechanisms.

Spherical motors in general, produce the desired unlimited spherical motions by utilizing electromagnetism or piezoelectricity. Kumagai et al. [4] and Kasashima et al. [5] both present electromagnetic motors which generally have a spherical rotor (made of a permanent magnet or a ferrous metal) and an electromagnetic stator. This arrangement yields the desired spherical motions through the generation of thrust forces on the rotor. These motors are generally complex in structure and have a slow response. On the other hand, piezoelectric spherical motors [6, 7] drive a spherical rotor with a traveling wave produced when voltage is applied to piezoelectric materials with different phases on the odd and even sets. Piezoelectric motors have a fast response with no electromagnetic interference, but suffer from small power output and low efficiency [8].

Mechanisms that produce spherical motions can be grouped without loss of generality as gimbal-type, parallel-linkage type, devices that change a sphere’s center of gravity, and sphere surface contact type devices. Osborne et al. [9] and Gao et al. [10] propose gimbal-type devices that generally possess a large workspace but are prone to the gimbal-lock and suffer from low stiffness and force output capacity. Parallel-linkage type devices [11, 12], and [13] feature high force bandwidths, low inertia, and passive back-driveability but have workspace boundary singularities and thus are unable to generate unlimited spherical motions. On the other hand, continuously changing the center of gravity of a sphere can result in unlimited spherical motions. Zhao et al. [14] and Gambari et al. [15] achieve this change in a sphere’s center of gravity by controlling the pose of pendulums within the sphere. However, the resulting devices are relatively hard to control and complex in structure.

Furthermore, Ip et al. [16] propose a surface contact type device that uses a complex mechanism and magnets to apply surface-contact forces on a sphere. This device is able to generate unlimited spherical motions, however it strives to eliminate workspace boundary singularities through reconfigurable control which is prone to errors. The Atlas motion platform [17] is another surface-contact type device that uses three omni-directional special wheels to drive a sphere and generate singularity-free unlimited spherical motions. It is easy to control, but has sphere contact discontinuities, is relatively expensive and cannot be easily scaled up or down due to its special wheels.

In this paper, we present the Hbot, a novel three degrees of freedom (DOF) spherical parallel mechanism for haptic simulations. It is able to produce unlimited spherical motions without workspace boundary or interior singularities which can be avoided completely at the design stage. The Hbot uses at least two steered and driven non-holonomic wheels to produce holonomic and unlimited spherical motions. Moreover, using more than two wheels results in a redundant mechanism which presents several other advantages such as improved stability. The Hbot features a small footprint, high force bandwidths, high stiffness and passive back-drivability. In addition, since we use standard non-holonomic wheels, the Hbot has relatively better traction with continuous sphere-wheel contact. It is relatively cheaper and easily scalable. A special case of our device occurs when steered and driven conventional tyres are used to produce holonomic motions on a plane [18].

We make the following contributions:

  • We present the kinematic equations of motion for the kinematic interface between a sphere and arbitrarily positioned, steered and driven non-holonomic wheels.

  • We conduct a general singularity analysis of this mechanism and show that workspace-boundary and workspace-interior singularities can be avoided at the design stage.

  • We implement a physical prototype of a spherical haptic interface with two steered and driven non-holonomic wheels.

  • We analyse the Hbot in terms of its kinematic and dynamic performance.

  • We evaluate the performance of the Hbot for rotational stiffness rendering and trajectory tracking.

Notation - Throughout this paper, angles are represented using Greek letters, vectors are in bold face, a hat denotes a unit vector, denotes the rotation matrix of frame with respect to frame , denotes the position vector from point to point , represents the velocity of point with respect to frame , represents the angular velocity of frame with respect to frame , and denotes the time derivative of vector in frame .

The rest of the paper is organized as follows; in section II, the kinematic analysis for the generalized problem is presented, in section III, the detailed singularity analysis is shown. Furthermore, in section IV, we describe an implementation of the proposed device and in section V, we conduct kinematic and dynamic performance evaluation for the prototype. Section VI details results for trajectory tracking and impedance control using the prototype and section VII then concludes the paper.

Ii Kinematic Analysis

The generalized problem is that of a sphere actuated by driven and steered standard wheels positioned at arbitrary contact points with arbitrary orientations. The kinematics problem then becomes finding a geometrically feasible configuration for the wheels such that there is no kinematic slip between the standard wheels and the sphere. In Fig. 1, we define the inertial frame to be at the center of the sphere with radius . Point is the contact point between a standard wheel of radius and the sphere where the subscript denotes a specific standard wheel, is the tangent plane to the sphere at . We then write the rotation matrix of the wheel frame with respect to the inertial frame as;

(1)

Where , , and are auxiliary frames required to define a set of simple rotations between the inertial frame and the wheel frame. is an arbitrary unit vector representing the steering axis of an arbitrary wheel, is the steering angle of the wheel connected to the link , is the cone angle which defines the plane containing circle on which point lies, and defines the driving angle of the wheel. is defined as a rotation about axis by an amount of . It can be obtained by using Euler parameters. Given that the unit vector can be expressed as , one can then obtain the relevant Euler parameters.

is the rotation matrix between auxiliary frames and defined as a rotation about (or ) by a constant cone angle , is the rotation matrix between auxiliary frames and defined as a rotation about axis (or ) by an amount of a constant angle , and is the rotation matrix between the auxiliary frame and the wheel frame defined as a rotation about the driving axis (or ) with an amount of . Having defined the relationship between the relevant frames, one can then write the non-holonomic constraint equations for an arbitrary wheel in contact with a sphere at a single point as;

(2)
(3)

Eq. 2 and Eq. 3, state that the velocity of point on the wheel side, with respect to the inertial frame and the velocity of point on the sphere side with respect to the inertial frame are equal in the axial () and tangential () directions of the wheel. The velocity , is then written as;

(4)

Where is the position vector from to the point at the center of wheel , and is the position vector from the center of wheel to the contact point , is the angular velocity of frame with respect to frame and is the angular velocity of frame with respect to the inertial frame . Next, the velocity can be expressed as;

(5)

where is the angular velocity of the sphere with respect to the inertial frame, and is the position vector from point to point on the sphere. Then the no-slip constraint equations become:

(6)
(7)

It is immediately straightforward to observe that the relevant position vectors and angular velocities of Eq. 6 and Eq. 7 are as shown below:

Thereafter, defining the angular velocity of the sphere with respect to the inertial frame as , these constraint equations become:

(8)

Or simplifying a bit further, we get:

(9)
Figure 2: Workspace-boundary singularity configuration

We make the following definitions:

and re-write Eqn. 9 as:

(10)

Having obtained two equations that describe the kinematic slip constraints for one wheel, it is then possible to generalize the solution and show that for arbitrary wheels, the constraint equations can be written as;

(11)

With Eqn. 11 above, the motion level kinematic analysis is complete and the configuration level kinematic analysis becomes straightforward as it involves finding the orientation of the sphere with respect to the inertial frame () given the orientation of the wheels or vice-versa. More specifically, if the orientation of the sphere with respect to the inertial frame is defined using a unit quaternion , then using the so-called quaternion propagation and integrating appropriately, the configuration level kinematics problem is readily solved resulting in a holonomic spherical mechanism. The quaternion propagation is shown in Eq. 12 and Eq. 13, and it relates the time derivative of the unit quaternion and the body angular velocity .

(12)
(13)

where is the skew-symmetric operator.

Figure 3: Workspace-interior singularity configurations

Iii Singularity Analysis

Parallel manipulators are prone to singularities and there are generally two types; workspace interior and workspace boundary singularities. Recall Eq. 11, the general non-holonomic constraint equation describing the kinematic interface between a sphere and arbitrary steered and driven wheels. Workspace interior singularities occur when is not full rank while workspace boundary singularities occur when is singular.

Iii-a Workspace Boundary Singularities

Workspace boundary singularities are based on matrix which is a block diagonal matrix in the general case of wheels. Since is diagonal, its determinant is the product of the determinants of corresponding to each wheel ;

(14)

The determinant of for each wheel can be found as;

(15)

Eq. 15 should then be set to zero to find the relevant singularity conditions. Given that and are non-zero for a realistic implementation, it then follows that is the required condition that yields workspace boundary singularities. A physical interpretation of this result states that workspace boundary singularities occur when the steering axis and the driving axis are aligned for a wheel such that velocities produced by steering the wheel and those produced by driving the wheel are linearly dependent. This singularity configuration is depicted in Fig 2 and can clearly be avoided at the design stage of a physical prototype.

Iii-B Workspace Interior Singularities

Workspace interior singularities are based on matrix which is a matrix in the general case of n wheels. Notice that is full rank in the case of only one wheel. This is clearly as a result of the fact that is a rotation matrix whose determinant is unity, and the matrix multiplying has linearly independent rows and columns. However, for two wheels (A & B), the matrix is expressed as:

(16)

Clearly, for this case the matrix loses rank only when . This implies that workspace interior singularities for the two-wheel case occur when the wheels are at the same orientation with respect to the sphere. Meaning that they are either collocated or one wheel is located at the antipodal point of the other wheel on the sphere. Fig. 3 shows the workspace interior singularity configurations for two wheels.

Intuitively, these singular configurations occur when the steering and driving velocities of one wheel are linearly dependent on those of another wheel. It is obvious that just like the workspace-boundary singularities, workspace-interior singular configurations can also be avoided at the design stage.

In order to fully control the orientation of the sphere in the manifold , one requires at least two active wheels. Therefore, in the case of n-wheels (), driving a sphere, it is sufficient to ensure that there exist at least a pair of wheels that produce linearly independent steering and driving velocities in order to avoid workspace interior singularities. Taking into consideration the fact that the proposed device can produce unlimited spherical motions, it follows that the singularity-free workspace covers the manifold .

Figure 4: 3D CAD rendering (left) and prototype (right)

Iv Implementation and Characterization

Iv-a Physical Implementation

A prototype of the spherical robot is designed using two wheels and as shown in Fig. 4. A ball-bearing cage is used to ensure proper sphere centering and to carry the weight of the end-effector which is a hollow plastic sphere of mass 467 and radius 102 . An important part of the design is the avoidance of singularities within the spherical unlimited workspace. Workspace boundary singularities are avoided by ensuring that the steering axis and the driving axis never become parallel for any given wheel. Workspace interior singularities are avoided by ensuring that the wheels can never collocated on the sphere such that they can always generate linearly independent steering and driving velocities. A singularity-free workspace is then achieved by selecting the design parameters as shown in Table I for the prototype herein.

Parameter Wheel A Wheel B
0.1745 0.1745
0 0
Table I: Design Parameters for the Prototype

Notice that the upper half of the prototype is left open by design to facilitate better interactions with the spherical end-effector. Furthermore, low-inertia direct drive actuators are used for the grounded motors which are 90 brushed motors while the actuators on the rotating links are low-weight (24 ) 3 motors with 17:1 gear reduction ratio. This specific selection of the actuators results in a prototype that both has a low apparent end-effector inertia and is passively backdriveable.

Iv-B Experimental Characterization

The bandwidth is an important property of force-feedback devices. The Hbot is a parallel mechanism with low-apparent inertia and direct drive actuation hence a high bandwidth is expected. Under closed loop orientation control, Fig. 5 is experimentally obtained and a bandwidth of 5.50 Hz can be observed.

Figure 5: Angle versus frequency plot under closed loop position control

Furthermore, since safety during operation is always important, the Hbot, a force-feedback device should be passively back-driveable. This problem is tackled at the design stage with the selection of direct drive actuators. We then verify passive backdriveability by measuring the minimum torque required to induce motion along the degrees of freedom of the device. Our results show that minimum torques of 5 mNm, 4 mNm, and 4 mNm are required to induce visible rotations about ,   and respectively.

Torques generated by the Hbot are expected to cover torque limits for specific haptic applications. The maximum instantaneous and continuous torques generated by the motors are Nm and Nm respectively and when mapped to the end-effector, they become Nm and Nm respectively. In Table II, we summarize results of our experimental characterization.

Criterion
Peak Inst. Torque[Nm] 1.4 0.94 0.84
Peak Cont. Torque[Nm] 0.11 0.15 0.065
Back-driveability [Nm]
Position Bandwidth [Hz] 5.50 5.50 5.50
Table II: Experimental Characterization of the Hbot

V Kinematic and Dynamic Performance

Figure 6: Inverse kinematics control to obtain joint variables at sampled points in

Here, we investigate the kinematic/dynamic performance of our prototype. Performance measures are generally required for the design, task planning and synthesis of manipulators. Optimal task-specific mechanism structures are obtained based on optimizing the desired performance metric. There are numerous measures of performance that exist in the literature which can be classified as kinematic/dynamic or local/global without loss of generality.

V-a Kinematic Performance

Kinematic performance metrics are based on the Jacobian matrix. They include but are not limited to the kinematic manipulability index [19] and the condition number [20] which are local indices (configuration dependent) but can be integrated over the mechanism’s workspace to generate global performance measures. The condition number has been proven to be a better measure of the degree of ill-conditioning of a mechanism than the manipulability index and hence the condition number which is an unbounded measure is adopted here. It is given as ;

(17)

Where J is the kinematic Jacobian and ||.|| denotes a frame-invariant norm given as;

with , where is the identity matrix and is the dimension of the Jacobian matrix. However, is unbounded and can result in computational problems. The reciprocal of - the local conditioning index, is then selected ;

(18)

A corresponding global performance measure - the global conditioning index, [21] demonstrating the distribution of the conditioning number over the entire workspace is adopted and is given as;

(19)

where are workspace nodes of the mechanism.

V-B Dynamic Performance

Dynamic performance metrics are based on the inertia matrix which is obtained here by Lagrange’s formulation. A widely used measure - the dynamic manipulability index , estimates the ability of a mechanism to generate acceleration based on a driving force at the joints and is selected here. It is given for non-redundant manipulators [22] as;

(20)

where is the inertia matrix. However, is unbounded so the normalized manipulability index is selected and given as;

(21)

where is the manipulability index at a given point and is the maximum manipulability index in the entire workspace. is then integrated over the entire workspace to generate the global manipulability index, as follows;

(22)

where are workspace nodes of the mechanism.

V-C Uniform Sampling of SO(3)

To calculate the performance measures enlisted in the preceding sections, knowledge of the mechanism’s workspace is paramount. The workspace in question is the 3D rotation group , and a uniform set of points within the workspace should be generated in order to ensure that the global performance metrics are calculated from evenly distributed sampled data points. Yershova et al. [23], use Hopf coordinates (, , ) to generate quaternions that parametrize . Each quaternion, can be written as a function of the Hopf coordinates as follows:

(23)

where , , have ranges of , , respectively. A set of uniformly sampled points in SO(3) is then generated by varying each coordinate between its limits with a specified step size .

V-D Inverse Kinematics Control and Performance Results

Upon generating the uniformly sampled set of orientations in the workspace, a set of joint variables corresponding to those points is then sought in order to calculate the desired metrics. Obtaining these points is however not straightforward because of the non-holonomic relationship between the end-effector pose and joint variables . Inverse kinematics control shown in Fig. 6 is then proposed to take the end-effector in simulation, from a fixed initial orientation to each of the sampled orientations on SO(3) such that the corresponding joint variables at those sampled points are calculated. In this control architecture, is the desired sampled orientation and is the corresponding calculated joint variable vector. is the control gain usually selected as a diagonal matrix. The quaternion error is formed from and as follows;

(24)

With this control scheme, results show RMS tracking error of rad, and percent RMS value with respect to the maximum position of . The performance metrics are then computed since both the sampled orientation in and the corresponding joint variable vector are now available. The was found to be while the normalized was found as . These values indicate an average performance for the Hbot. However, one can find design parameters that optimize a desired performance metric.

Vi Impedance Control in SO(3) and Experimental Evaluation

Impedance control is a well-established control strategy for the interactions between a robot’s end-effector and the environment. Modeling the prototype as an impedance-type device, open-loop impedance control is then an appropriate strategy for trajectory tracking and virtual stiffness rendering. In the task space, minimal representations are typically used to represent the orientation of the end-effector and this leads to representation singularities. Moreover, this minimal representation does not hold a physical meaning considering the definition of an end-effector’s rotational impedance.

Caccavale et al. [24], propose proper global representations to define the rotational impedance of a robot’s end-effector. Specifically, using unit quaternions and to represent the orientation of the actual and desired end-effector frames respectively, the impedance equation can be written as:

(25)

Where and are the angular velocities of the desired and actual frames with respect to the inertial frames respectively, is the contact moment at the end-effector, , , and are positive definite matrices describing the generalized inertia, rotational damping and rotational stiffness respectively. is expressed as;

(26)

Where,

(27)
(28)
(29)

and is the skew symmetric operator. To evaluate the trajectory tracking performance of the prototype, a chirp orientation signal with a 35 degree amplitude and frequencies increasing up to 1 Hz is applied as the reference orientation trajectory. This signal is applied along rotational axis and Fig. 7 shows the chirp signal tracking performance along this axis. The RMS trajectory tracking error is calculated to be 1 % when the errors are generated at the joints. However, this is not realistic as problems such as slip, stiction, improper centering tend to increase tracking errors for the prototype. Thus, an external error measurement system is required. Here an external camera is used to track the orientation of the end-effector by aligning the axis of rotation with the axis of the camera such that the tracked point is sure to follow a circular path on a plane. Fig. 7 also shows the results obtained from the external tracking system where the RMS tracking error was found to be less than 8%.

Figure 7: Trajectory tracking about axis

Furthermore, virtual rotational stiffness levels around 2 different axis have been rendered. Fig 8 presents torque-deflection data measured for these renderings after applying known torques to the sphere and measuring its deflection. Table III presents values characterizing the quality of the line fit, the slope of these lines, and the rendering error. The implication of this result is that even under open loop impedance control (without force feedback), high fidelity impedance renderings with RMS errors less than 5.5% can be achieved with the device. A video of the prototype can be found at https://youtu.be/R8WMv_JjPUM

{tabu}

to 0.4 | X[c] | X[c] | X[c] | X[c] | X[c] | Axis & Desired stiffness [N.m/rad] & Slope of fitted line [N.m/rad] & Rendering error [%] & Quality of fit ()

& 4.00 & 3.80 & 5.00 & 99.36

& 6.00 & 5.72 & 4.67 & 98.13

& 2.00 & 1.89& 5.50 & 99.42

& 5.00 & 4.74& 5.2 & 99.42

Table III: Spring rendering results for the device
Figure 8: Virtual rotational stiffness rendering about and directions

Vii Conclusion and Future Work

In this paper, we presented the kinematics, singularity analysis, functional prototype and real-time impedance control of the Hbot - a novel 3 DoF spherical parallel mechanism. For the first time, the general kinematics of a sphere actuated by steered and driven non-holonomic wheels has been explored. A major result obtained here is the fact that singularities can be avoided at the design stage. Workspace-boundary singularities are avoided by ensuring that the driving and steering axis of each wheel are never parallel while workspace-interior singularities are avoided by ensuring that for at least two wheels in the general case of -wheels, the steering and driving velocities of one wheel are linearly independent of those of the other wheel.

Furthermore, the Hbot features parallel kinematics with analytical solutions and is modeled as an impedance-type device. As a result of its direct-drive and grounded actuation, open-loop impedance control results show the possibility of using the device for haptic simulations. It is noteworthy that proper end-effector (sphere) centering is crucial to achieve good results. Ongoing work include optimal dimensional synthesis of the device and experimental evaluation of its effectiveness in various applications from flight simulation to neurophysiology studies.

Acknowledgment

The author would like to thank Volkan Patoglu for very useful discussions.

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