Bearing structure for a robot speed reducer, actuator for a robot and robot

The warehouse structure for robot speed reducers uses sloped ball bearings with lightweight clamping elements and seals to address the weight and power issues of cross-roller bearings, improving efficiency and rigidity while preventing lubricant leakage.

DE112022007300T5Pending Publication Date: 2025-04-03FANUC LTD
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Patent Information

Application Number
DE112022007300
Authority / Receiving Office
DE · DE
Patent Type
Applications
Current Assignee / Owner
Filing Date
2022-09-15
Publication Date
2025-04-03

AI Technical Summary

Technical Problem

Cross-roller bearings used in robot actuators are heavy and have low output power due to their large roller frame, while slanted ball bearings, although lighter, lack the rigidity and sealing properties of cross-roller bearings.

Method used

A warehouse structure for a robot speed reducer using sloped ball bearings with clamping elements made of lightweight materials and seals to maintain rigidity and prevent lubricant leakage, combined with a pre-load mechanism to enhance alignment and reduce friction.

Benefits of technology

The proposed structure achieves improved output efficiency and reduced weight of the speed reducer by using lightweight clamping elements and effective seals, maintaining rigidity and preventing lubricant loss, thus enhancing the overall performance of the actuator.

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Abstract

A bearing structure for a speed reducer for a robot, an actuator for a robot, and a robot are described, wherein the stiffness of an actuator incorporating a bearing is equal to that of a crossed roller bearing, and lubricating oil used inside the speed reducer does not leak to the outside. This bearing structure comprises a bearing capable of absorbing a moment acting in any direction, a plurality of first clamping elements for axially clamping an outer ring of the bearing between them, and a first sealing element located where the plurality of first clamping elements face each other. At least one section of the first clamping elements is made of a material with a lower specific density than steel.
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Description

AREA

[0001] The present disclosure relates to a bearing structure for a speed reducer of a robot, an actuator for the robot, and the robot. BACKGROUND

[0002] A robot can change the position and posture of a work tool by driving a component, such as an arm. The robot is equipped with a drive unit that includes an electric motor for moving the component. For example, if the robot has a jointed part, the drive unit for moving the component is positioned at the joint. The drive unit has a power transmission mechanism for transmitting a rotational force from one element to another.

[0003] A power transmission mechanism typically includes a speed reducer for amplifying the torque of a motor. A shaft reduction gear with a crossed roller bearing is known as a speed reducer. Furthermore, technologies for reducing the weight of such a reducer have been proposed (see, for example, Patent Literatures 1 and 2).

[0004] In addition, since a crossed roller bearing has high rotational friction, a technique is known in which an angular contact ball bearing is used as a bearing and aluminum, etc., is used for the components of the bearing to reduce the weight of the bearing (see, for example, Patent Literature 3). [CITATION LIST][PATENT LITERATURE] [PTL 1] JP 2019-219041 A [PTL 2] JP 2020-050242 A [PTL 3] JP 2010-090942 A SUMMARY[TECHNICAL PROBLEM]

[0005] Because a crossed roller bearing has a compact shape and can withstand a relatively large tilting moment of the arm, it is often used as a bearing to support the output shaft of a speed reducer for a robot. However, a crossed roller bearing is heavy and has large rolling friction, so the output power of the speed reducer tends to be low. On the other hand, when using a bearing other than a crossed roller bearing, such as an angular contact ball bearing, it is desirable that the rigidity of an actuator incorporating the bearing be equivalent to that of the crossed roller bearing, and that the lubricating oil used in the speed reducer does not leak to the outside. [SOLVING THE PROBLEM]

[0006] One aspect of the present disclosure is a bearing structure for a speed reducer of a robot, the bearing structure comprising: a bearing capable of receiving a moment acting in an arbitrary direction; a plurality of first clamping members configured to sandwich an outer ring of the bearing in an axial direction; and a first sealing member provided at a position where the plurality of first clamping members oppose each other, wherein at least a portion of the first clamping member is configured from a material having a lower specific gravity than steel.

[0007] Another aspect of the present disclosure is an actuator for a robot comprising: the above bearing structure; and an electric motor for driving an output shaft of the speed reducer.

[0008] Another aspect of the present disclosure is a robot comprising the above bearing structure or actuator. BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a schematic perspective view of a robot according to an embodiment. Fig. Figure 2 is a schematic cross-sectional view of a bearing structure capable of supporting the robot of Fig. 1 to be raised. Fig. Figure 3 is a schematic radial cross-sectional view of a speed reducer. Fig. 4 is a partially enlarged view of Fig. 2. Fig. 5 is a view showing a modification of Fig. 4 shows. Fig. 6 is a schematic cross-sectional view of a bearing structure according to a comparative example. DESCRIPTION OF EMBODIMENTS

[0009] Fig. 1 is a schematic perspective view of a robot 10 according to a preferred embodiment. For example, the robot 10 is an industrial articulated robot and has a base 12 fixed to an installation surface, such as a factory floor, a rotating body 14 attached to the base 12 for rotation about an axis J1, an upper arm 16 attached to the rotating body 14 for rotation about an axis J2, a lower arm 18 attached to the upper arm 16 for rotation about an axis J3, a wrist 20 attached to the lower arm 18 for rotation about an axis J4, and a flange 22 attached to the wrist 20 for rotation about an axis J6. The flange 22 is provided with a work tool (not shown) corresponding to an operation to be performed by the robot.

[0010] Although the robot 10 according to the embodiment is a vertical articulated robot with six drive axes, the present disclosure is not limited thereto, and any robot that can change its position and posture by any mechanism may be used.

[0011] Fig. 2 is a cross-sectional view showing a structural example of a robot actuator 24 including a bearing structure for a speed reducer of the robot 10. Here, a structural example for rotating the lower arm 18 around the axis J4 will be described. However, the present example is not limited to this and can also be applied to a structure around, for example, the axis J1, J2, J3, J5, or J6. The actuator 24 is arranged on the side of the lower arm 18 opposite the side on which the wrist 20 is arranged.

[0012] The actuator 24 has a bearing structure described below and an electric motor 30 comprising a rotor 26 and a stator 28, with the rotor 26 being attached to a shaft 32. The shaft 32 acts as the output shaft of the electric motor 30 and rotates about the axis J4. The shaft 32 is supported by bearings 46, 48 attached to a housing 44, and a rotational force of the shaft 32 is transmitted to a flange 42 via a speed reducer 34.

[0013] A resin protective tube 33 is disposed inside the shaft 32. The protective tube 33 is formed into a cylindrical shape along an inner surface of the shaft 32, and a supply member, such as an electric wire, an air hose, or an optical communication cable, is inserted into the protective tube 33. The protective tube 33 allows the supply member to be disposed inside the joint of the robot 10.

[0014] Fig. 3 shows a schematic radial cross-section of the speed reducer 34. The speed reducer 34 in the illustrated example is a stress wave reducer and has a wave-generating element 36 to which a rotational force is applied, an elastic cylindrical element 38 arranged radially outwardly of the wave-generating element 36, and an annular element (also referred to as a circular spline) 40 arranged radially outwardly of the cylindrical element. The wave-generating element 36 includes a hub 37 having an elliptical shape when viewed in the axial direction and a ball bearing 39 arranged on an outer peripheral surface of the hub. The cylindrical element 38 has teeth 38a on its outer peripheral surface, is configured to elastically deform with the rotation of the hub 37, and is connected to the hub 37 by a coupling interface such as a ball bearing. B. a bolt 84, attached to the housing 44.The annular member 40 has teeth 40a on its inner peripheral surface which mesh with the teeth 38a of the cylindrical member 38, and outputs a rotational force reduced in speed according to a reduction ratio corresponding to the number of teeth of the cylindrical member 38 and the number of teeth of the annular member 40.

[0015] On the lateral side of the annular member 40, a main bearing capable of supporting a moment acting in any direction is arranged. The main bearing in the illustrated example is a combination bearing consisting of two adjacent angular contact ball bearings 50a and 50b, but a tapered roller bearing, etc., can also be used as a combination bearing. When the ball bearing is used, the rotational friction of the bearing can be reduced compared to a crossed roller bearing, etc., thereby reducing torque loss due to friction and improving the output efficiency of the reduction gear. Instead of the combination bearing, a double-row bearing or a crossed roller bearing with no interface for mounting to other components can also be used.

[0016] Fig. 4 is a partially enlarged view of the bearings 50a and 50b. The bearing 50a has an outer ring 52a and an inner ring 54a, and the bearing 50b has an outer ring 52b and an inner ring 54b. The outer rings 52a and 52b are clamped by at least two first clamping members 56 and 58, and the first clamping members 56 and 58 are fastened to each other by a bolt 60 acting as a clamping functional part. Furthermore, the first clamping members 56 and 58 have shoulders 57 and 59 that contact end surfaces of the outer rings 52a and 52b, respectively.

[0017] The inner ring 54a of the bearing 50a and the inner ring 54b of the bearing 50b are clamped by at least two second clamping elements 68 and 70, which are fastened to each other by a bolt 72 or a similar fastener acting as a clamping function. Furthermore, the second clamping elements 68 and 70 have shoulders 69 and 71 that contact the end surfaces of the inner rings 54a and 54b, respectively. The arrangement of the shoulders on the second clamping elements necessarily achieves a function of aligning the axes of the bearings and the clamping elements.

[0018] At least one of the clamping elements 56, 58, 68, and 70 is made of a material with a lower specific gravity than iron or steel. Specific examples of such materials include titanium, aluminum, magnesium, and plastic, etc. By manufacturing the clamping elements from a material lighter than iron, the weight of the bearing structure can be reduced, and a speed reducer with high response performance for an energy-saving robot can be obtained.

[0019] Among the clamping elements 56, 58, 68, and 70, for the clamping element 68 on which the circular gear 40 is mounted on the inner ring side of the bearing, deformation of the clamping element 68 will result in deformation of the circular gear 40, which may ultimately cause vibration in the speed reducer. Therefore, it may be advantageous to use iron or steel as the material for the clamping element 68 instead of a lightweight material such as aluminum.

[0020] Generally, the lighter the bearing structure, the lower its rigidity. Therefore, in the present embodiment, a plurality of clamping members are used to clamp the outer ring or the inner ring of the bearing as described above, and these are fastened using the axial force of the bolt to eliminate or reduce the gap in the axial direction. Furthermore, since the bearing structure according to the present embodiment is for a robot reducer and has an open-type bearing, a seal is provided between the members contacting the outer ring to prevent leakage of lubricating oil, etc., which is filled inside the speed reducer and lubricates the bearing. A specific example will be described below.

[0021] A part “A” in Fig. 4 shows an example of a canceling mechanism for canceling an axial gap between the axial end surface of the bearing (outer ring) when the outer rings 52a and 52b are clamped, and the axial end surface of the clamping member that contacts the axial end surface of the bearing. Specifically, at least one of the clamping members 56 and 58 (in this case, the clamping member 56) has a leg 74 at a portion that contacts the clamping member 58, and the leg 74 contacts the clamping member 58 to form an axial gap 76 between the clamping members 56 and 58. The width of the leg 74, that is, the radial length (or seating area), is set to a size such that a surface of the clamping member 58 that contacts the leg 74 is not depressed.

[0022] By tightening the bolt 60 from this state, the leg 74 fits onto the clamp member 58, and the gap 76 at a non-fitting portion becomes smaller due to the elastic deformation of the clamp member. As a result, the axial gap between the bearing end surface and the clamp member end surface that contacts the bearing end surface is eliminated, thereby generating an anti-slip frictional force on the bearing end surface, and the outer ring does not rotate relative to the clamp member. In addition, the bearing itself is clamped in the axial direction and elastically deforms radially outward, so that an anti-slip frictional force is also generated on the cylindrical mating surface, suppressing rotation of the outer ring. By suppressing rotation in this way, fretting wear on the clamp member side can be prevented.

[0023] When the leg 74 is provided, the clamping elements 56 and 58 can complete preload adjustment for the bearing simply by tightening the bolt 60 to a certain torque. On the other hand, when the leg 74 is not provided, it is possible to adjust the preload of the bearing itself by tightening the bolt 60. However, in addition to the fact that dimensional control of the parts becomes complicated and costs increase, caution is required because the axial force of the bolt becomes small and the bolt 60 may loosen.

[0024] A first sealing member 62 is provided at a portion where the clamping members 56 and 58 face each other (specifically, touch or are close to each other), thereby preventing the lubricating oil, etc., used in the speed reducer from leaking to the outside. The first sealing member 62 is, for example, an O-ring and is made of an elastically deformable material. When the above-mentioned axial gap decreases due to elastic deformation, the first sealing member 62 is compressed, and the first sealing member 62 elastically deforms to follow the axial gap. Furthermore, it is desirable that the first sealing member 62 be designed and configured so that it does not generate permanent deformation even in the most compressed state.With regard to the sealing structure, the first sealing part 62 is preferably arranged in the area between the outer rings 52a, 52b of the bearing and the clamping bolt 60.

[0025] At least a portion of the clamping member 56 that contacts the outer ring 52a is configured to hold a second sealing member 64 that seals between the clamping member 56 and the rotating output member (in this case, the annular member 40) of the speed reducer 34. The second sealing member 64 is, for example, an oil seal. By using a low-friction oil seal as the second sealing member 64, it is possible to further reduce the torque loss in the speed reducer 34 due to rotational friction and improve the output efficiency of the speed reducer 34.

[0026] At least a portion of the clamping element 58 that contacts the outer ring 52b contacts the component of the speed reducer 34 (in this case, the cylindrical element 38) and has a third sealing part 66 that seals at the contact point between the clamping element 58 and the cylindrical element 38. The third sealing part 66 is, for example, an elastically deformable O-ring. With regard to the sealing structure, the third sealing part 66 is preferably arranged at a location on the clamping elements 56, 58 clamping the outer rings 52a, 52b that protrudes radially inward from a shoulder portion 59 of the element 58 that contacts the reducer 34.

[0027] The gap on the inner ring of the bearing can also be eliminated by a similar structure to that of the outer ring. A part “B” of Fig. 4 shows an example of a canceling mechanism for canceling an axial gap between the axial end surface of the bearing (inner ring) when the inner rings 54a and 54b are clamped, and the axial end surface of the clamping member that contacts the axial end surface of the bearing. Specifically, the clamping member 68 or 70 (in this case, the clamping member 70) has a leg 78 at a portion that contacts the clamping member 68, and the leg 78 contacts the clamping member 68 to form an axial gap 80 between the clamping members 68 and 70. The width of the legs 78, that is, the radial length (or seating area), is set to a size such that a surface of the clamping member 68 that contacts the leg 78 is not depressed.

[0028] By tightening the bolt 72 from this state, the leg 78 fits onto the clamp member 68, and the gap 80 at a non-fitting portion becomes smaller due to the elastic deformation of the clamp member. As a result, the axial gap between the bearing end surface and the clamp member end surface that contacts the bearing end surface is eliminated, thereby generating an anti-slip frictional force on the bearing end surface, and the inner ring does not rotate relative to the clamp member. In addition, the bearing itself is clamped in the axial direction and elastically deforms radially outward, so that an anti-slip frictional force is also generated on the cylindrical mating surface, suppressing rotation of the inner ring. By suppressing rotation in this way, fretting wear on the clamp member side can be prevented.

[0029] When the leg 78 is provided, the clamping elements 68 and 70 can complete the preload adjustment for the bearing simply by tightening the bolt 72 to a certain torque. On the other hand, when the leg 78 is not provided, it is possible to adjust the preload of the bearing itself by tightening the bolt 72. However, in addition to the fact that dimensional control of the parts becomes complicated and costs increase, caution is required because the axial force of the bolt becomes small and the bolt 72 may loosen.

[0030] As described above, for example, the bolt can be used as the clamping functional part that secures the clamping elements for clamping the outer or inner ring of the bearing. However, since the clamping elements are made of aluminum, etc., in order to reduce their weight, it is preferable to use a relatively large number (e.g., 8 to 16) of relatively small bolts (e.g., M4 or smaller) to tighten the clamping elements at equal intervals (in the circumferential direction) to suppress local deformation of the clamping element. By using a plurality of small bolts, it is also possible to prevent the screw seat surface from being countersunk, which is more likely to occur with lightweight materials.

[0031] As in Fig. As shown in Figure 4, a washer 86 can be used as a canceling mechanism for canceling the axial gap between the end face of the bearing 50a during clamping and the end face of the clamping member 56 that contacts the end face of the bearing 50a. Similarly, a washer 88 can be used as a canceling mechanism for canceling the axial gap between the end face of the bearing 50b during clamping and the end face of the clamping member 58 that contacts the end face of the bearing 50b. For example, a plurality of washers with different thicknesses can be prepared in advance, and the axial gap can be measured so that a washer that fills the measured gap can be selected and inserted to eliminate the gap.

[0032] Washers 86 and 88 may be made of iron, steel, stainless steel, copper, etc., and are preferably made of a relatively hard material. However, a wave washer and other structurally elastically deformable materials are not preferable, as they may reduce the rigidity of the bearing structure.

[0033] Fig. 5 shows a modification of Fig. 4. The embodiment of Fig. 5 differs from that of Fig. 4 in that the part “A” (ie the section in which the first clamping elements 56 and 58 are in contact with each other or close to each other) of the Fig. 4 of that of Fig. 4 is shifted towards the flange 42 (to the left). In particular, in the example of Fig. 4 the part “A” adjacent to the outer ring of the bearing (in this case an outer peripheral surface of the outer ring 52a), while in the example of Fig. 5 the part “A” is located further towards the flange 42 (opposite the adjacent outer ring 52b) than the axial end surface of the outer ring 52a. The other parts in Fig. 5 can be the same as in Fig. 4.

[0034] Specific embodiments in which the first clamping element is provided with the shoulder contacting the end face of the outer ring include an embodiment in which substantially one element 56 has the shoulder 57, as shown in Fig. 4, as well as an embodiment in which the plurality of elements 56, 58 cooperate to form the shoulder 57, as in Fig. 5. The same applies to an embodiment in which the second clamping element is provided with a shoulder that contacts the end surface of the inner ring.

[0035] Fig. 6 shows a comparative example in which a crossed roller bearing is used as the main bearing. Among the Fig. 6 are the parts that correspond to the Fig. 2 or Fig. 4 are provided with reference numerals calculated by adding 100 to the reference numerals in Fig. 2 or Fig. 4, and detailed explanations are omitted.

[0036] An actuator 124, as in Fig. 6, has a crossed roller bearing 151 as the main bearing. The crossed roller bearing 151 has an outer ring 153 and an inner ring 155, and the inner ring 155 is fixed to a housing 142 by a fastener 149, such as a bolt.

[0037] Generally, a crossed roller bearing is often used as a bearing to support the output shaft of a robot reducer because the crossed roller bearing has a compact shape and can withstand a large tilting moment of a robot arm. However, the crossed roller bearing is essentially an iron block and has the disadvantage of being heavy. In particular, as shown in Fig. 6, the weight becomes even greater if the outer ring 153 or the inner ring 155 is made large and a fastening interface (bolt 149) for other components is provided thereon.

[0038] The above-mentioned tension shaft reducer is often used in small robots, and in most cases, a crossed roller bearing is used in the tension shaft reducer. Currently, the crossed roller bearing accounts for about 60% of the weight of the speed reducer. Furthermore, the larger the tension shaft reducer becomes, the more the weight of the crossed roller bearing becomes a bottleneck, and the weight tends to increase rapidly relative to the size of the speed reducer. In addition, the crossed roller bearing has high rolling friction, which affects the output efficiency of the speed reducer.

[0039] In contrast, in the present embodiment, the plurality of clamping members are used to clamp either the outer ring or the inner ring or both, and at least one of the clamping members is made of a material lighter than steel, thereby making the bearing structure lighter, and by providing the sealing part between the clamping members, the bearing structure that can be used as a speed reducer for a robot is provided.

[0040] If you compare the combination of the Fig. 2 with a crossed roller bearing in which the outer and inner rings are made of steel, and the bearing part maintains the same moment stiffness, reducing the weight of the clamping element reduces the moment stiffness of the lightweight part compared to steel, resulting in a reduction in the moment stiffness of the actuator as a whole. Therefore, by using the above-mentioned clamping element and the gap cancellation mechanism to make the moment stiffness of the bearing part with preload adjustment higher than that of a crossed roller bearing, for example, the actuator as a whole can have the same or greater stiffness even if the moment stiffness of the clamping element is reduced.

[0041] As in Fig.As shown in Figure 2, the clamping elements 68, 70 that clamp the inner rings 54a, 54b have a coupling interface 82 with another component (in this case, the housing 42) that forms the robot 10. Similarly, the clamping elements 56, 58 that clamp the outer rings 52a, 52b have a coupling interface 84 with another component (in this case, the housing 44) that forms the robot 10. Bolts or similar fastening means, for example, can be used as the coupling interfaces 82 and 84. In the present embodiment, the outer ring or the inner ring of the bearing is thus not provided with a coupling interface with other components, and the coupling interface is provided in the clamping element.Therefore, as long as the outer ring and the inner ring are not provided with a coupling interface, it is also possible to use a crossed roller bearing in the present embodiment, and the bearing can be made into a crossed roller bearing while achieving a reduction in the overall weight.

[0042] According to the above-mentioned embodiment, in the bearing structure including the bearing supporting the output shaft of the speed reducer for the robot, the rigidity and sealing properties around the bearing can be ensured to function as a bearing, and the output efficiency of the speed reducer can be improved by reducing the weight of the clamping member. Furthermore, the electric motor and the bearing structure for the speed reducer do not necessarily have to be an integrated unit (e.g., an actuator unit). For example, the speed reducer, the above-mentioned bearing for the speed reducer, and the electric motor can be directly integrated into a robot arm formed as a cast part. LIST OF REFERENCE SYMBOLS 10 robots 12 Base 14 rotating body 16 upper arm 18 Forearm 20 wrist 22 flange 24 Actuator 30 electric motor 32 Wave 34 speed reducer 50a, 50b ball bearings 52a, 52b outer ring 54a, 54b inner ring 56, 58 first clamping element 60, 72, 82, 84 bolts 62 first sealing part 64 second sealing part 66 third sealing part 68, 70 second clamping element 74, 78 thighs 86, 88 washer 151 crossed roller bearings QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited patent literature

[0000] JP 2019-219041 A

[0004] JP 2020-050242 A

[0004] JP 2010-090942 A

[0004]

Claims

[1] Bearing structure for a speed reducer of a robot, the bearing structure comprising: a bearing capable of supporting a moment acting in any direction; a plurality of first clamping elements configured to sandwich an outer ring of the bearing in an axial direction; and a first sealing part provided at a position where the plurality of first clamping elements are opposed to each other, wherein at least a portion of the first clamping member is configured from a material having a lower specific gravity than steel. [2] The bearing structure of claim 1, further comprising a plurality of second clamping members configured to sandwich an inner ring of the bearing in an axial direction, wherein at least a portion of the second clamping member is configured from a material having a lower specific gravity than steel. [3] A bearing structure according to claim 2, wherein the first or second clamping member has a shoulder portion contacting an end surface of the outer ring or the inner ring of the bearing. [4] The bearing structure according to claim 2 or 3, wherein at least one of the first and second clamping members includes a canceling mechanism configured to cancel an axial gap between an axial end surface of the outer ring or the inner ring and an axial end surface of the clamping member contacting the axial end surface of the outer ring or the inner ring. [5] A bearing structure according to any one of claims 1 to 4, wherein at least a portion of the first clamping member includes a second sealing member that seals a rotating output member of the speed reducer. [6] A bearing structure according to any one of claims 1 to 5, wherein at least a portion of the first clamping member contacts a component of the speed reducer and includes a third sealing member sealing between the first clamping member and the component at a contact point therebetween. [7] A bearing structure according to any one of claims 1 to 6, wherein the bearing is of an open type and is configured to be lubricated by lubricating oil filled inside the speed reducer. [8] A bearing structure according to claim 2 or 3, wherein the first or second clamping member has a coupling interface for coupling with another component constituting the robot. [9] A bearing structure according to claim 2 or 3, wherein the plurality of first clamping members or the plurality of second clamping members are fixed to each other by a bolt. [10] A bearing structure according to any one of claims 1 to 9, wherein the bearing is a paired duplex angular contact ball bearing. [11] A bearing structure according to any one of claims 1 to 10, wherein the bearing supports an output shaft of a stress wave reduction gear. [12] Actuator for a robot, comprising: the bearing structure according to any one of claims 1 to 11; and an electric motor for driving an output shaft of the speed reducer. [13] A robot comprising the support structure according to any one of claims 1 to 11. [14] A robot comprising the actuator according to claim 12.

Citation Information

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