A highly-underactuated robotic hand with force and joint angle sensors

A highly-underactuated robotic hand with force and joint angle sensors Long Wang*, Joseh DelPreto**, Sam Bhattacharyya*, Jonathan Weisz***, Peter K. Allen*** *Mechanical Engineering, **Electrical Engineering, and ***Comuter Science, Columbia University Abstract This aer describes a novel underactuated robotic hand design. The hand is highly underactuated as it contains three fingers with three joints each controlled by a single motor. One of the fingers ( thumb ) can also be rotated about the base of the hand, yielding a total of two controllable degrees-of-freedom. A key comonent of the design is the addition of osition and tactile sensors which rovide recise angle feedback and binary force feedback. Our mechanical design can be analyzed theoretically to redict contact forces as well as hand osition given a articular object shae Key words Robotic Hands Underactuation Tactile sensors I. INTRODUCTION Underactuated robotic hands have become quite oular over the last few years, for a number of reasons. Hands such as [, 2, 3, 4] occuy a niche among the wide sectrum of robotic hands that lie between simlistic 2-fingered industrial griers and comlex 5-fingered humanoid hands such as the shadow hand [5]. Utilizing usually one actuator or less to curl a single finger, these hands allow a much more simlified control scheme comared to traditional fully actuated multi-finger hands. The joint comliance and multi-fingered configurations in these under-actuated hands also gives them the beneficial roerty of being able to successfully gras comlex objects of relatively unknown shae and orientation (see [6]) as well as to be easily integrated into larger systems such as an arm to interact with the environment ([7]). Low cost, simle design, and the otential for mass marketability all make under-actuating hands quite romising for current and future develoment in robotic rostheses and humanoid robotics (for examle, see [8]). An oen question relating to these hands is just how much underactuation can be tolerated before the hand loses its ability to gras and maniulate objects. This aer discusses the design of the Columbia Hand, which is a highly under-actuated hand, containing 3 fingers with 3-DOF each, and only two actuators: one for closing the fingers and one for rotating the thumb around the base. To sulement the lack of controllable DOF s, a rich set of sensors has been included that can estimate ) the joint angles and 2) tactile contacts on each segment of the hand. While some research has been done regarding the integration of tactile sensors with the rincile of underactuation (e.g. [9, 0]), very little work has been done to integrate both osition and tactile feedback. Such feedback, combined with the adatability inherent to underactuation, allows for sohisticated control algorithms and enhanced gras quality of a wide variety of objects. This work was suorted in art by the National Science Foundation under Grant IIS-09044. II. PRINCIPLES OF TP-UA MECHANISMS As shown in Fig., the Tendon-Pin (TP) Underactuated (UA) mechanism was integrated on each joint of each finger. This design integrates small segments of a robotic finger via in joints and a tendon, allowing for a robust design which is both easily controlled and inherently adatable to a wide variety of objects. The resonse of this TP-UA when closing on an object can be divided into three essential stages: the initial stage, the re-shaing stage, and the closing stage. A. Initial Stage In the initial stage, shown in Fig., the finger is straightened by return srings. There is a tendon going through holes on every segment, with one end fixed to the distal segment and the other connected to the lifting mechanism actuator. As shown in Fig., the gras rocess is temorarily considered with only the first two DOFs for convenience. (c) Fig..The rinciles of a tendon-in under-actuated finger, illustrating the initial stage, the re-shaing stage, and (c) the closing stage. Notes: the distal segment; 2 the middle segment; 3 the fixed segment; 4 the lifting mechanism; 5 the return sring; 6 the joint in; 7 the tendon; 8 the grased object B. Pre-Shaing Stage The re-shaing stage will be considered as the interval beginning when actuation is alied and ending when any segment touches the object. During this time, the finger acquires a re-shaed bend. As shown in Fig., once the actuator moves down and alies a tensile force on the tendon, the finger will start closing. All the joints will be rotating simultaneously in a couled relationshi. The secifications of this rocess and couled relationshi deend on design arameters such as tendon osition (distance from joint) and the return sring, which will be discussed below. C. Closing Stage The closing stage will describe the interval beginning with the moment of object contact and ending when no segment

can continue to move (when the gras is comleted). As shown in Fig. (c), if the middle segment is blocked and the tendon is continuously ulled down, the distal segment can continue to bend the two joint angles have been decouled by the object. In this way, the TP-UA finger can comlete a gras task with a single actuator. There are several advantages: ) Small volume and weight due to the simle mechanism, esecially comared to link and gear style UA mechanisms; 2) Reduced number of required lifting mechanisms. Secifically, this aer rooses a closing system of 9 DOFs actuated by a single lifting mechanism. 3) Increased anthroomorhism. Due to its mimicry of human muscles, the re-shaing rocess of a tendon-based design naturally reflects human movement (for examle, see [, 6]). This transmission will be discussed in detail below. III. FINGER MODEL & MECHANICAL ANALYSIS To effectively aly the TPUA in robotic fingers, a feasible design must be considered with all arameters otimized regarding geometry, mechanics, and motion lanning. While the mechanics of a linkage-driven self-adative finger is discussed in [2], this analysis demonstrates the mechanics of a tendon-based adative underactuated hand. As shown in Fig. 2 and, a mechanical analysis is addressed in detail for each of the three stages introduced above. A. Assumtions, Parameters, and Symbol Definition The symbols used in this section are defined in TABLE I. According to Fig. 2, there are four imortant design arameters secifying this model, including L 0, L, R 0, and K. Fig. 2. The mechanical analysis of the re-shaing stage Notes: the distal segment; 2 the middle segment; 3 the fixed segment; 4 the lifting mechanism; 5 the return sring; 6 the joint in; 7 the tendon; 8 the grased object There are several assumtions alied to construct this model. First, the finger mechanism itself is assumed to be frictionless; thus, every contact oint of the tendon can be modeled as a smooth ulley (shown in Fig. 2 ). Note that this model does, however, account for friction between the finger and the object. Second, the finger comonents are assumed to be massless; thus, there are no gravitational effects as the joints rotate. Third, the model alies symmetric and identical segments. Finally, all finger movements will be considered as quasi-static rocesses. TABLE I Symbol Definition Symbol Descrition Position L 0 The height of each segment. Fig. 2 L The height of closer tendon turning. Fig. 2 R 0 The width between tendon holes and joints. Fig. 2 K The stiffness coefficient of return srings. Fig. 2 θ i The rotation angle of the ith joint. Fig. 2 δ i The ith tendon turning angle. Fig. 2 T 0 The alied force on the tendon roduced by the lifting mechanism. Fig. 2 The force alied on the sto ball roduced by Fig. 2 T T 2, T 2, and T 22 r r 2, r 2 and r 22 M Si F i the tendon. The force alied on tendon turnings of segments roduced by the tendon. The lever arm of the moment with resect to the first joint roduced by T. The lever arms of the moment with resect to the st and 2 nd joint roduced by T 2, T 2 and T 22. The torque roduced by the return sring, alied at the ith joint toward the ith segment. The reaction force alied on the ith segment roduced by the object this force includes the effects of friction. B. Analysis during the Pre-Shaing Stage Fig. 2 shows the mechanical analysis of the re-shaing stage, during which the joint angles are couled. The mathematical derivation below investigates the exact nature of this couling given the above design arameters, one can determine the unique joint angle couling rocess. The following results can be roved geometrically assuming the symmetric configuration described above: 2 2, 3 4 () 2 2 r R0, r2 l sin( / 2) (2) r2 l0 sin( 0 2 / 2), r22 l sin( 3 / 2) 2 2 where l L R, tan ( L / R ), i 0, (3) i i 0 i i 0 Fig. 2 Fig. 2 Fig. 2 Fig. 2 k i The lever arm of F i. n i The direction normal to the finger surface at the contact oint of the ith segment. τ i The direction tangential to the finger surface at the contact oint of the ith segment; N i The rojection of F i onto the n i direction, which equals the normal force roduced by the object. f i The rojection of F i onto the τ i direction, which equals the friction roduced on contact surfaces. ψ i The angle between f i and F i, which indicates the friction angle (rad); But note that if the design is not symmetric, the following are still valid:, s. t. ;, s. t. ( ) 2 2 3 2 4 3 4 2 where η i can be a design arameter ranging from 0 to. Second, consider force equilibrium, assuming quasi-static rocesses. According to the force balance at each tendon turning oint, the following results can be derived: T T, T 2T sin, T 2T sin, T 2T sin (4) 0 2 0 2 0 2 22 0 3 Third, consider moment equilibrium, assuming quasi-static rocesses. According to the moment at each joint, the

following results can be derived: T r T2 r2 M S, M S M S0 K (5) T2 r2 T22 r22 M S 2 M S, M S 2 M S 20 K2 M S0, M S20 are re-loading torques alied on the corresonding joints (due to stretching of the srings in the initial stage), and M S, M S2 are due to additional stretching as the finger rotates. Summarizing () through (5), it can be concluded that MS0 K T0 P( ) (6) R 2l sin( / 2)sin( / 4) 0 K sin sin( ) Q( ) (7) 2 4 2 ( ) 2 2 2 lp where M S 0 K l0 Q( ) sin sin( 0 ) 2 lp( ) l 2 4 Thus, given the design arameters, one can calculate the relationshi between θ and θ 2 (Fig. 3b), i.e. how the two angles are couled, by substituting θ into (6) and solving (7). In addition, it is imortant to note that the tendon force T 0 (the force the actuator must suly) is a function of only θ. Fig. 3. The relationshi between θ and θ 2, demonstrating the ability to redict the re-shaing osition as well as a strong resemblance to human movement. simulates the re-shaing while shows how the two angles are couled. Both analyses are done with the same arameter configuration as the secified design which will be discussed later, and the re-loading torque is set to zero for convenience. From Fig. 3, it can be concluded that a natural re-shaing rocess similar to a human hand is obtained during this stage. C. Analysis during the Closing Stage Now consider the geometry and mechanics during the closing stage, as shown in. Unlike the re-shaing stage, the two angles θ and θ 2 are not couled. Instead, they are now decouled by the shae and contact force of the grased object. Thus the relevant investigation of this stage regards the determination of the gras force given a articular grased object, i.e. given how θ and θ 2 are distributed, one can comute the gras force. First, note that the geometric issue is the same as in the re-shaing stage, which means that equations (), (2), and (3) can still be used in this section. Second, the force equilibrium is the same as in the re-shaing stage with the excetion of the contact forces, so equation (4) remains valid. The contact forces are the suerosition of normal and frictional forces, which deend on the coefficient of friction and the object s shae. Here, the directions are assumed to be known for the articular object. Third, the moment equilibrium is similar to that of the revious stage. Thus, (5) can be modified to obtain. The mechanical analysis of the closing stage T r T2 r2 F k M S( ) T2 r2 T22 r22 F2 k2 M S2( 2) M S( ) which can be rewritten as F k A( T0, ) F k 0 A or F 2 k 2 B ( T 0,, 2) F 2 k 0 2 B (9, 0) where the functions A and B can be determined from (8). From equation (9), it can be seen that F can be determined by the tendon force T 0, the distal segment angle θ, and the lever arm k ; also F 2 can be determined by T 0, θ, θ 2, and k 2. The contact force transmission characteristic of this UA mechanism is analyzed below. By fixing the contact oints and lotting the two contact forces over all sets of angle combinations, Fig. 5 can be obtained. According to the results of Fig. 5, the transmission ratio F i /T 0 remains relatively high as the two angles change, and in articular F /T 0 remains relatively constant during the closing stage. However, F 2 is very small when the middle segment angle is near zero, which is accetable since the middle segment will bend during most gras tasks. IV. DESIGN OF ACTUATION SYSTEM Notes: the labels are the same as those in Fig. 2. Fig. 5.The force transmission characteristic Notes: in this figure, the contact oint is fixed, k =k 2=(L +L 0)/2; and all the other design arameters are the same as the secified design which will be discussed later; the range of both angles is from 0 to π/2. The Columbia Hand constitutes three fingers integrated on a alm, one of which is able to rotate around the wrist (acting as a thumb). Instead of alying three lifting mechanisms (one for each finger), there is one mechanism for all three fingers. While various methods of actuation exist, such as a lanetary gear system [3], this article rooses a movable T (8)

block mechanism driven by a single linear actuator. This allows all unconstrained fingers to move even if one finger is externally locked. The mechanism accomlishes this by creating arallel kinematic chains which allow common sets of tendon wire to be shared by all three fingers. A. Princile of the Movable Block mechanism The movable block mechanism shown in Fig. 6 was roosed to rovide tendon force and vertical dislacement for the three fingers. This section analyzes this system in detail, utilizing techniques similar to those described in [6]. Fig. 6. The movable block mechanism, which allows all three fingers to be controlled by a single motor but still conform to an arbitrary object shae. Notes: the movable ulley; 2 the rigid frame; T kj e.g. T b the tendon force roduced by the blue tendon in the figure; T the force alied on the frame (roduced by the lead-screw mechanism). Though the analysis in the revious section considered one wire within each finger, the actual actuation mechanism in our design includes a air of wires running symmetrically through each figure. Each finger has two wires attached to the distal halanx, but each of the other two ends of these wires is attached to the distal halanx of one of the other fingers. The wires are routed around a set of ulleys in order to minimize friction in the system. Thus, each finger is kinematically couled with the other two fingers, and forcing one finger to extend would cause the other two fingers to close. Fig. 6 yields the following kinematic constraints: () (2) where d Σ is the distance traveled by the linear actuator, Δl j,k is the length change of k color tendon within the jth finger. Also, the following force constraints can be obtained: (3) (4) where T j,k is the tension of the k color wire in the jth finger. Now consider the total force exerted by one articular finger on an object during a gras: F k, x, k F ˆ ˆ ˆ k Fk, y [ n, k n2, k n3, k ] 2, k T (5) k, j j F kz, 3, k where is the unit normal of the ith segment contact for the kth finger, and η i,k is the transmission ratio (F i /T 0 ) of the for the ith finger contact and the kth finger. Recall equation (4,5); η i,k is indeendent of T 0. The normal and transmission ratio matrices can be combined into a single Jacobian matrix J, which is a function of the geometry: F k = J k ( Σ j T k,j ) (6) The seudo-inverse can then be taken of the Jacobian to solve for the sum of tension in the wires in that finger: ( ) (7), ( ) (8) Substituting these results into to equation (4), T T J F (9) j,2,3 kb, m, g j, k,2,3 Consider a change in the force alied by the linear actuator, T {( J ) F ( J ) F} (20) Assuming no change in geometry, as would be the case for a closed gras, (20) simlifies to: T ( J ) F (2) This demonstrates that the force alied by the motor is distributed across the fingers according to their geometry. Particularly, with a constant force alied by the actuator, i.e. ΔT Σ = 0, the external forces alied by the fingers remain couled and thus may still change according to (2). Thus, one benefit is that an external force alied to one finger increases the tension in that finger. Because the tensions are couled, the disturbance on that finger will result in comensation forces alied by the other fingers. With the aroriate grasing geometry, this means that a disturbance force on a grased object actually results in a tighter gras. B. Design and Integration of the Actuation System As shown in Fig. 7, the actuation system includes the lead-screw transmission connected to the motor and the tendons routed throughout the alm. The lead-screw transmission rovides a non-backdrivable characteristic, allowing for accurate ositioning. The actuation system also includes the control of thumb rotation. Here, a worm-gear mechanism is utilized to rovide non-backdrivable abduction and adduction of the thumb. Fig. 7.Actuation system design featuring a single lead-screw mechanism for the tendons and a non-backdrivable motor for the thumb rotation. Notes: the thumb;2 alm;3 the lifting mechanism;4 the lead screw transmission;5 the wrist frame V. FINGER DESIGN & SENSOR INTEGRATION The Columbia Hand successfully integrates joint angle sensors and force sensors on each finger segment. Secifically, there are ten joint angle sensors (nine for finger joints and one for the thumb rotation) and nine force sensors (one for each segment). It is imortant to integrate sensors into a UA hand. Angle sensors allow one to achieve osition feedback and therefore gain knowledge of object shae, while force sensors allow one to obtain contact forces. These measurements can

rovide substantial information regarding object shae as well as gras quality, which are vital for the successful maniulation of an object. Such sensory integration therefore adds the benefits of osition feedback often obtained in fully actuated hands. A. The Finger Design with Sensor Integration As shown in Fig. 8, ball bearings are emloyed in each joint. Each finger extends 53 mm, and the design arameters defined reviously are set as L 0 =36mm, L =2mm, R 0 =4mm. Every joint utilizes a rotary otentiometer as an angle sensor, which has several advantages. Fig. 8.Finger design with sensors Notes: force sensor;2 angle sensor;3 ball bearing; 4 thrust bearing; 5 hex-shaft First, otentiometers offer a very linear characteristic and high accuracy (as demonstrated in the following section). Second, they are very comact, inexensive, and straightforward to imlement. Third, comared to Hall Effect sensors, otentiometers are smaller and will not be affected by metal comonents like bearings and return srings or magnetic roerties of grased objects. Also, each segment has a convex curve equied with a Force Sensitive Resistor (FSR), which is covered by a gri ad. While other research features caacitive tactile sensors [0] or iezoelectric olymer film elements [9], FSRs rovide an inexensive, sensitive, simle alternative. The gri ads are relatively soft and therefore conform to objects. This, in combination with a high coefficient of friction, hels distribute forces over the FSRs entire surface area and increases the quality of grass. The FSR circuitry constitutes a simle voltage divider. While the FSR is a nonlinear device which can be highly sensitive to changes in low forces but much less sensitive to changes at high forces, the nonlinear transfer characteristic of the voltage divider hels comensate by inducing greater outut changes at smaller values of the sensor s resistance. For the angle sensors, however, a non-inverting o am configuration is emloyed with the otentiometer as the feedback resistor. This roduces a linearly changing voltage in resonse to a linearly changing resistance. B. Sensor Performance Currently, the force sensors are only used for binary data (whether a segment has contacted the object); if a segment s FSR roduces a voltage above a redetermined reference, it will indicate contact. This data is sufficient to substantially increase gras quality and control (see [9]). The angle sensors, however, must rovide recise information for osition feedback. They are tested as below, with ground truth obtained with a MicroScribe. In this test, one joint is set to several ositions while data is recorded from the MicroScribe, voltage outut, and Matlab. Fig. 9 indicates a good linear characteristic of the circuitry while Fig. 9 indicates a high overall osition feedback accuracy. Fig. 9. The voltage outut and angle determined by Matlab is lotted versus the actual angle. High correlation coefficients as well as a sloe very close to indicate the osition feedback system s high degree of accuracy. VI. TESTS The rototye of the Columbia Hand is shown as Fig. 0. Fig. 0. The comlete design of Columbia Hand, featuring angle and tactile sensors as well as a TPUA mechanism driven by a single motor. It has three fingers, each of which has three segments, and a rotational DOF at the base of the thumb. There are two actuators, one controlling the 9 DOFs of the fingers and the other controlling the thumb s osition. There are ten angle sensors and nine force sensors. This data is received by a DAQ board and rocessed via Simulink. A. The Pre-shaing Test Fig. comares lots generated by Matlab based on the angle sensors in one finger as it closes without an object with ictures of the actual finger during the same test. (c) Fig.. Results of the re-shaing test with an individual finger: shows the animation comuted from sensory feedback while shows the ground truths; (c) comares the first, middle, and last ositions of and. These demonstrate a high degree of accuracy of the osition feedback system and a natural reshaing rocess strongly resembling the rediction above. It can be concluded from the results that i) the finger has a relatively large worksace; ii) Fig. (c) indicates the accurate real-time tracking rovided by the angle sensors; iii) the test results differ slightly from the redictions of the revious analysis (Fig. 3). These discreancies can be attributed to gravity as well as friction; resistance in the joints delays the rotation of lower joints, causing the distal segment to rotate substantially before the others begin to move, creating the observed differences. The second test is similar to the first test, but investigates

the re-shaing of the whole hand. As shown in Fig. 2, ictures are continuously taken and the osition feedback is resented via Matlab. It indicates that the movement is very natural and stable and that the osition feedback is accurate. Fig. 2.The results of re-shaing stage of the whole hand. The right-hand ictures show the animation generated by Matlab using the sensory feedback, while the left-hand ictures show the ground truths. These results demonstrate a very human-like characteristic. B. The Gras Test In this section, two objects are grased by the Columbia Hand while sensory feedback is recorded: a cylindrical CD case and a toy horse. In the animations roduced by Matlab, a red segment indicates contact on that surface with the grased object. These results are consistent with the ground truth next to each animation. The results are shown as Fig. 3. Grasing a CD case Grasing a toy horse Fig. 3 The Columbia Hand successfully grasing objects. The right column shows lots generated by Matlab from sensor data; red segments indicate contact with the object. These demonstrate stable grass for differently shaed objects and the ability to redict gras quality based on sensory data. It can be concluded from the results that i) the Columbia hand can successfully adat to different shaes; ii) the angle sensors are successfully integrated and rovide recise ositional information; iii) the force sensors are sensitive and work well as logic indicators for whether or not contact has been achieved. From this data, one can obtain knowledge about object shae and gras quality before attemting to maniulate the object. Such integration therefore facilitates the oeration of underactuated hands in unknown environments or situations which require stable grass; it combines many benefits offered by full actuation with the simlicity and adatability of underactuation. VII. FUTURE WORK AND CONCLUSION There are several interesting issues which can be considered in future research. First, the re-shaing rocess can be otimized by modifying the design arameters, so that the worksace of each finger and variety of grasable objects can be maximized. Second, a control law can be considered to control the gras rocess with feedback information from the sensors, thereby utilizing the angle and force information to enhance gras quality and control. Similarly, the force sensors can be used to rovide analog force values rather than binary ones this can consequently rovide more thorough information regarding gras quality as well as object roerties. Benchmark tests can also be erformed to facilitate comarison of this hand with other underactuated hands, as roosed in [4]. Finally, the hand will also be mounted as the end effector of a robotic arm, allowing the hand to oerate as art of a larger system and interact with the surrounding environment. This aer roosed a novel underactuated robotic hand design the Columbia Hand. It has three fingers, which are imlemented with tendon in underactuation mechanisms. The three fingers are actuated by only two motors, one of which controls the thumb rotation. The Columbia Hand is equied with angle and force sensors, which can rovide recise angle feedback and sensitive binary force feedback. Furthermore, this style of imlementation rovides a system which can be analyzed theoretically to redict contact forces as well as hand osition given a articular object shae. Tests indicate that the Columbia Hand is functional at gras tasks for a variety of objects. Additionally, its ractical feedback allows for gras quality analysis and object recognition as well as imroved control algorithms that can comensate for intrinsic error. Such develoments demonstrate the ower of underactuated hands and their growing otential for humanoid robotics. REFERENCES [] A.M. Dollar and R. D. 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