Design of All-Accelerometer Inertial Measurement Unit for Tremor Sensing in Hand-Held Microsurgical Instrument

Design of All-Accelerometer Inertial Measurement Unit for Tremor Sensing in Hand-held Microsurgical Instrument

Wei Tech Ang, Pradeep K. Khosla, and Cameron N. Riviere The Robotics Institute Carnegie Mellon University Pittsburgh, PA 15213 USA

Abstract – We present the design of an all-accelerometer equipped with a standard IMU, containing three miniature inertial measurement unit (IMU). The IMU forms part of an rate gyros (Tokin CG-16D) and three accelerometers intelligent hand-held microsurgical instrument that senses its (Crossbow CXL02LF3 tri-axial), at the distal end of the own motion, distinguishes between hand tremor and instrument handle from the instrument tip. intended motion, and compensates in real-time the erroneous The key problem of this conventional IMU design is motion. The new IMU design consists of three miniature that the orientation estimation depends totally on the noisy dual-axis accelerometers, two of which are housed in a sensor suite at the distal end of the instrument handle, and low-cost gyros. The noisy gyros harm the instrument tip one located at the proximal end close to the instrument tip. displacement estimation in two ways: (i) Any angle By taking the difference between the accelerometer readings, estimation errors at the distal end are translated into a we decouple the inertial and gravitational accelerations from magnified displacement error at the instrument tip; (ii) the rotation-induced (centripetal and tangential) Errors in orientation estimation cause errors in gravity accelerations, hence simplifies the kinematic computation of compensation in the accelerometer readings and in turn angular motions. We have shown that the error variance of corrupt the tip displacement estimation. the Euler orientation parameters θx, θy and θz is inversely High quality gyros with the precision needed for this proportional to the square of the distance between the three application are too bulky to be fitted into a hand-held sensor locations. Comparing with a conventional three gyros and three accelerometers IMU, the proposed design reduces device [5, 6], and they are almost always too costly [7]. the standard deviation of the estimates of translational On the other hand, inexpensive, batch-processed gyros, displacements by 29.3% in each principal axis and those of although having suitable size and weight, currently lack the Euler orientation parameters θx, θy and θz by 99.1%, the level of required precision. Technology breakthrough 99.1% and 92.8% respectively. in the gyro manufacturing process is not likely in the near future due to challenges associated with micro- I. INTRODUCTION miniaturization of gyros [8]. Exploiting the more affordable and mature micro-machined accelerometer In microsurgery, surgeons must perform manipulation tasks technology, researchers have proposed gyro-free or all- that approach the limits of human positioning accuracy. accelerometer sensor designs. Chen et al. [9] proposed an The envelope of human manipulation precision is bounded original cube configuration design with six by involuntary hand movements. The best-known accelerometers, the minimum number of sensors to erroneous hand motion is physiological tremor. Tremor is recover all the six kinematic parameters. However, no defined as any involuntary, approximately rhythmic, and comparison with the conventional three gyros three roughly sinusoidal movement [1]. In ophthalmological accelerometers design has been reported. microsurgery, its significant component is found to be an We propose in this paper a novel IMU design using oscillation at 8-12 Hz whose frequency is independent of three dual-axis accelerometers and no gyros. Section II the mechanical properties of the hand and arm [1], and with gives a brief overview of the intelligent microsurgical amplitude of about 50 µm peak-to-peak or less [2, 3]. instrument to provide more insight to the background of These inherent limitations complicate many delicate the problem. It also describes the design of the proposed surgical procedures, and make some types of intervention new inertial measurement unit and discusses the impossible. Examples of delicate procedures include motivations for choosing this configuration. Section III removal of membranes as thin as 20 µm from the retina, derives the motion equations of the instrument from a and operating on tumors in close proximity to crucial brain differential kinematics perspective that decouples tissues. rotation-induced accelerations from inertial accelerations In previous work we have developed an intelligent and gravity induced accelerations. In Section IV, error hand-held microsurgical instrument that senses its own covariance analysis is performed on the proposed design, motion, distinguishes tremor from the intended motion of and compared to the standard three-gyro, three- the surgeon, and deflects the instrument tip in realtime accelerometer configuration. with an equal but opposite motion in order to compensate the erroneous movement [4]. The first prototype is II. DESIGN OF INERTIAL MEASUREMENT UNIT accelerometer is located at the front end close to the intraocular shaft, measuring motion in X and Y directions. The intelligent microsurgical instrument is made up of There are a few reasons for adopting this design for three sub-systems: the sensing system, the filtering the sensing system. Firstly, in our application of sensing algorithms, and the tip manipulator system. An overview and compensating erroneous hand movement during of the system is shown in Fig. 1. microsurgery, the internally referenced inertial sensing technology emerges to be the most suited. From a Sensing System technical standpoint, it meets all mandatory technical specifications (resolution and accuracy: < 1 µm, system Motion of Inertial bandwidth: > 15 Hz). More importantly, from an end-user instrument Measurement Unit perspective, all externally referenced sensors (E.g. Optotrak and Polaris, Northern Digital Inc.; Fastrak, Polhemus, Ultratrak, Polhemus Inc.; Constellation, Intersense Inc.), require a line of sight or other similar Filtering Differential Kinematics Compute motion of tip provision and may be a hindrance to the way the surgery Algorithms - Estimation of w.r.t. local frame is performed, given the tight workspace the surgeon has to

undesired work with. Moreover, in our attempt to sense and motion compensate hand tremor, the signal of interest becomes the higher frequency hand tremor, while the intended Inverse kinematics motion becomes the “noise,” as it were. When the filtering algorithm separates the slower varying voluntary motion from the tremor, the low-frequency sensor drifts are

Controller removed at the same time. As a result, precise tracking of the absolute pose of the instrument is not of importance, as long as we can remove gravity from the kinematic model to recover the tremor motion. Parallel Compensation of Manipulator tremor ZB

Manipulator System

Fig.1 System overview of the intelligent microsurgical Front sensor instrument for tremor compensation suite

The inertial measurement unit senses the motion of the instrument and feeds the raw sensory information to the filtering algorithms. The filtering algorithms remove the intended motion, compensate for the phase shift due to Dual-axis analog and digital filters, and model and estimate the accelerometer tremor based on a dynamic sinusoidal model [10]. The motion of the instrument tip is then computed by forward kinematics after removal of the gravity. Next, the manipulator system deflects the tip in real-time in an opposite direction but equal magnitude to the motion due to hand tremor. Only the design of the inertial XB measurement unit will be discussed in this paper. The inertial measurement unit for our hand-held microsurgical instrument consists of a total of three miniature Analog Devices ADXL-202E dual-axis accelerometers (dimensions: 5 mm × 5mm × 2mm, weight: < 1g). These sensors are housed in two locations YB in the instrument handle, as shown in Fig. 2. The sensor Dual-axis suite at the back end of the instrument houses two dual- Back sensor accelerometers axis accelerometers in orthogonal orientations (two X-, suite one Y-, and one Z-sensing directions). One dual-axis Sensing direction

Fig. 2. Design of the inertial measurement unit. Secondly, miniature accelerometers are more superior body principle axes: XB, YB, and ZB. Fig. 3.1b shows the to miniature low-cost gyros, in terms of noise, linearity accelerometer locations, the sensing axes and kinematic and accuracy. Thirdly, compared to other existing all- frames of the instrument. accelerometer IMU in a cube configuration, the proposed The XB, YB and ZB axes of frame {B} are initialized to design can use off-the-shelf dual-axis accelerometers, coincide with XW, YW and ZW axes of the world coordinate since the sensing directions can be orthogonal. This system {W} at time t = 0. To derive the angular velocity B B B B T results in a more compact, simpler, and cheaper design. vector, Ω = [ ωx ωy ωz] from the accelerometers, we Fourthly, having six accelerometers provides one degree start from the motion equation of a rigid body in E3 space. B of sensing redundancy in each of the three translational The total accelerations, Ai, each accelerometer at {i} B DOF and enables the error covariances to be reduced by senses include the inertial acceleration of the body, ACG, techniques such as Kalman filtering. Last but not least, the the gravity, Bg, and the rotation-induced accelerations: the B most compelling reason to have an IMU design with centripetal acceleration, Ai/C, and the tangential B accelerometers placed apart from each other is that this acceleration, Ai/T, B B B B B configuration enables high quality sensing, to be shown in Ai = ACG + g + Ai/C + Ai/T , i = 1, 2, 3; (1a) the next sections. BA = BA + Bg + B Ω×B Ω × R+ Bα × R (1b) i CG 14244 4443 III. DIFFERENTIAL SENSING KINEMATICS rotation−induced We define a body frame {B} at the base of back sensor where the R is the vector from the unknown instantaneous suite, with its ZB-axis collinear with the instrument tip as center of rotation to the point of sensing. shown in Fig 3. Taking the difference between the accelerations readings at {1}, {2} and {3}, the non-rotation induced B B acceleration components ACG and g are eliminated, since ZB the linear inertial acceleration of the body and the gravity should be identical at different locations, {3} BA = BA – BA = B B BP + B BP X 13 3 1 Ω × Ω × 13 α × 13 3 = [Ba • •]T (2a) Y3 13x B B B B B B B B A23 = A3 – A2 = Ω × Ω × P23 + α × P23 P B T 13 = [• a23y •] (2b) P23 P3 B B B B B B B B Z1 A12 = A2 – A1 = Ω × Ω × P12 + α × P12 B T = [• • a12z] (2c) {1} X B 1 Z2 where the symbol • denoted undefined quantity and Pij = P B B B T 12 [ pijx pijy pijz] , i, j = 1, 2, 3, is position vector from {i} P1 P2 to {j} with respect to {B}, which are known values from XB {B} {2} Y2 the hardware design and calibrations. The differential velocity at the location {i} at time t is given by integrating Y Zw B (2), B B B Uij(t) = ∫ Aij(t) dt + Uij(t – T), i, j = 1, 2, 3; (3) Yw where T is the sampling period.

Xw {W} Taking only the defined dimensions in (2a-c) and Fig. 3 Kinematic representation of microsurgical instrument solving the simultaneous equations yield the angular velocity vector, B –1 B B B T In the back sensor suite, there are two dual axis ΩA(t) = PD UD = [ ωAx ωAy ωAz] , (4) B B B T B B accelerometers, A1 = [ a1x • a1z] and A2 = [• a2y B T B B B where a1z] . The third dual axis accelerometers, A3 = [ a3x a3y T •] is located at the front sensor suite close to the  B B  B 0 p13z − p13y  u13x  instrument tip. The symbol • means undefined. The  B B   B  locations of the accelerometers with respect to the body PD = − p23z 0 p23x  ; U D =  u23y  frame {B} are known from the mechanical design and  B p − B p 0   B u   12 y 13x   12z  through sensor calibrations. Without loss of generality, we assume that the proof masses of each of the dual axis The angular displacement vector is then given by accelerometer are coincident at {1}, {2} and {3} BΘ(t) = ∫ BΩ(t) dt + BΘ(t – T) (5) respectively, and all the sensing axes are aligned with the The directional cosine matrix that relates the body where the tip position with respect to the body frame {B} B T frame {B} to the world coordinate system {W} at each is Ptip = [0 0 l1] . Integrating (14) we get the sampling interval is updated by first updating the instantaneous tip displacement quaternions, WP (t) = WU (t) dt + WP (t – T) (16) ~ tip ∫ tip tip q&(t) = Ω(t)q(t) (6)

q(t) = ∫ q&(t) dt + q(t – T) (7) IV. ERROR COVARIANCE COMPARISON OF IMU DESIGNS  0 ω z − ω y ω x    ~ 1 − ω 0 ω ω Ω =  z x y  (8) In this section, we show the superiority of the proposed 2  ω y − ω x 0 ω z    IMU design over the conventional three gyros three − ω x − ω y − ω z 0  accelerometers design. We do so by comparing the error covariance in the estimated angular and translational ~ where Ω is the augmented cross product matrix. In terms displacement of the instrument. of quaternions, the directional cosine matrix is Since we have two accelerometers in each sensing W CB = axis, compounding the redundant sensing information 2 2 2 2 q0 + q1 − q2 − q3 2(q1q2 − q0 q3 ) 2(q1q3 + q0 q2 )  results in an improvement of 29.3% or a factor of 2 in  2 2 2 2  the standard deviations of the translational displacement  2(q1q2 + q0 q3 ) q0 − q1 + q2 − q3 2(q2 q3 − q0 q1 )  measurements, assuming all accelerometers have identical  2(q q − q q ) 2(q q + q q ) q 2 − q 2 − q 2 + q 2   1 3 0 2 2 3 0 1 0 1 2 3  noise characteristics. (9) σ 2 σ The effective translational body motion with respect to σ = Ai = Ai , for σ 2 = σ 2 . (17) A 2 2 Ai Aj {W} is obtained from removing the gravity component σ Ai +σ Aj 2 from the accelerometer readings, To analyze the error variance of the angular W W B W AiE = CB Ai – g (10) displacement estimations, we first define the sensor W T T where i = 1, 2, or 3 and g = [0 0 -g] . Integrating (9), we vectors, G = [ωx ωy ωz] and A = [a1x a1z a2y a2z a3x T get the effective velocities at {i}, a3y] . Taking into account the stochasticity of the sensors, W W W the sensor vectors become UiE(t) = ∫ AiE(t) dt + UiE(t – T) (11) A = Ā + δA; G = G + δG (18) The effective velocity at the origin of {B} with W W W W T respect to {W}, UE = [ uEx uEy uEz] is found by where Ā and G are the means, δA and δG are zero mean transforming the position and angular velocity vectors into random vectors with known covariances, T {W}, C(A)6×6 = E[(A – Ā) (A – Ā) ], (19a) W W B PiB(t) = CB(t) (- Pi), (12) T W W B C(G)3×3 = E[(G – G )(G – G ) ]. (19b) Ω(t) = CB(t) Ω(t) (13) Using the variance of the Analog Devices ADXL-202 and then compound the results derived from different accelerometers (~ 3850 mm2/s4) obtained from empirical accelerometers by their respective error variances, W W W W W W data, and using T = 1 ms, the variance per sampling period uEx = ΣA13x( u1x + p1Bz ωAy – p1By ωAz) + of d , d and d are computed and tabulated in Table I. W W W W W x y z ΣA31x ( u3x + p3Bz ωAy – p3By ωAz) (14a) T The Euler angles, ΘG = [θGx θGy θGz] , from the gyros W W W W W W are obtained from direct integration of the sensed angular uEy = ΣA23y( u2y + p2Bx ωAz – p2Bz ωAx) + W W W W W velocity. For simplicity in comparison, we use a ΣA32y ( u3y + p3Bx ωAz – p3Bz ωAx) (14b) trapezoidal integration rule: W W W W W W uEz = ΣA12z( u1z + p1By ωAx – p1Bx ωAy) + ΘG = 0.5⋅T⋅ (G[t + 1] + G[t]) (20) W W W W W ΣA21z ( u2z + p2By ωAx – p2Bx ωAy) (14c) 2 C(ΘG ) = 0.25⋅T ⋅(C(G[t + 1]) + C(G[t])) (21) 2 2 2 where ΣAjid = σ Aid /(σ Aid +σ Ajd ), d = x, y, or z. Using the variance of the Tokin CG-16D gyros (~ 2.0 To obtain the instantaneous velocity the instrument deg2/s2) obtained from empirical data, and a sampling tip with respect {W}, time of T = 1 ms, the individual variance or the diagonal W W W W B Utip(t) = UE (t) + Ω (t)× CB(t) Ptip (15) elements of the covariance matrix C(ΘG) are computed and tabulated in Table I.

From Section III, vector ΘA and the accelerometer vector A are related by vector function f, ΘA = f (A) = f (Ā + δA) (22) ZB Taking the Taylor Series expansion [11], {3} f (Ā + δA) = f (Ā) + ∇f δA + ½∇2f δA2 + … (23) P(θAx, θAy) X where ∇if·δAi is the informal notation for the ith order term 3 in the multidimensional Taylor Series and ∇f is the

Jacobian of f evaluated at Ā. 4σθAx or 4σθAy Y 3  ∂f1 ∂f1 ∂f1   L  ∂a1x ∂a1y ∂a3x  P(θGx, θGy)  ∂f2 ∂f2 ∂f2  ∂f (A) L 4σθGx or 4σθGy ∇f = (A) = ∂a ∂a ∂a  (24) ∂A  1x 1y 3x   M M O M  BP  ∂f6 ∂f6 ∂f6  X 13  L  1 ∂a1x ∂a1y ∂a3x  A=A {B}

The covariance of ΘA up to the first order is then T Y2 C(ΘA ) ≈ ∇f C (A) (∇f) + … (25) Performing a design optimization, we arrivev at a B T B Fig. 4 Comparison of the joint distributions of θx and θy design configuration of P12 = [-d1 d1 0] , P23 = [d3 −d3 T B T estimates between an all-gyro (P(θGx, θGy)) and an all- d2] , and P13 = [−d3 d3 d2] . Using T = 1 ms, the variance accelerometer (P(θAx, θAy))design. Higher density regions at the per sampling period of θAx, θAy and θAz are centers represent higher probability.   2 2 −8  1 1  2 σ θAx = σ θAy = 2.76 ×10 + deg , (26a)  2 2  The standard deviation of the θAx, θAy, and θAz are  2d1 d 2  reduced by 99.1%, 99.1% and 92.8% over those of θGx,  2  θGy, and θGy respectively. The intuition of this result is 2 −8  1 1  d 2   2 σ = 2.76 ×10 +   deg ; (26b) depicted in Fig. 4, taking θGx and θGy as examples. The θAz  2    d3 2  d1d3  discs labeled P(θGx, θGy) and P(θAx, θAy) are the joint   T T distributions of the ΘGxy = [θGx θGy] and ΘAxy = [θAx θAy] Considering the physical constraints of a handheld estimates from gyros and accelerometers respectively. device, we design the sensor positions to be d1 = 20 mm, Higher density regions at the centers represent higher d2 = 100 mm, and d3 = 12 mm. With this design, the probability. With the same uncertainty distribution at {3}, B B results of (26a-b) are tabulated in Table I. the larger | P13| (= | P23|) gets, the smaller the angle sustaining the cone will become. This angle may be TABLE I viewed as being 4 times the standard deviation (99.9% Comparison of error variance and standard deviations of confidence level) of θAx or θAy, since σθAx and σθAy are translational and angular measurements by a conventional three identical. The noise in the θ measurement is the worst gyros three accelerometers (3G-3A) design versus the proposed Az because it would be impractical to increase the length all-accelerometers (6A) design. B | P12| due to the physical constraints in the handheld instrument. Measurement θ ,θ θ dx, dy, dz x y z On top of the 29.3% direct improvement in the Error 3G-3A 1 × 10-6 1 × 10-6 1.54 × 10-8 translation sensing, the improvement in the orientation Var., deg2 deg2 mm2 2 estimation will result in a much complete removal of the σ gravity, which in turn has a great impact on the quality of 6A 7.44 × 10-11 5.17 × 10-9 7.7 × 10-9 2 2 2 the translation sensing since gravity is typically two deg deg mm orders of magnitude larger than hand accelerations.

Error 3G-3A 1 × 10-3 1 × 10-3 1.24 × 10-4 Std. deg deg mm Dev., V. CONCLUSION σ 6A 8.63 × 10-6 7.19 × 10-5 8.77× 10-5 We have presented the design of an all-accelerometer deg deg mm inertial measurement unit for tremor sensing in a hand- % 99.1 % 92.8 % 29.3 % held microsurgical instrument. The inertial measuring Reduction unit consists of three miniature dual-axis accelerometers, housed in two locations. At the back sensor suite, we have two dual-axis accelerometers, with sensing axes: one [11] R. Smith, M. Self, and P. Cheeseman, “Estimating along BX-, one along BY-, and two along BZ axis (the long Uncertain Spatial Relationships in Robotics”, in axis of the instrument). One dual-axis accelerometer is Autonomous Robot Vehicles, Eds. I.J. Cox and G.T. located at the front end close to the instrument tip, Wilfong, Springer-Verlag, 1990, pp.167-193. measuring motion in BX and BY directions. With differential sensing, we are able to decouple the rotation- induced motion from the translational motion and the gravity. Comparing with a conventional three gyros and three accelerometers IMU, the proposed design reduces the standard deviation of the estimates of translational displacements by 29.3% in each principal axis and those of the Euler orientation parameters θx, θy and θz by 99.1%, 99.1% and 92.8% respectively.

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