Control method for three-degree-of-freedom articulated bodies for robots.

Abstract

An articulated body having three degrees of freedom for a robot, comprising a platform (2) and three motors (3a, 3b, 3c), each connected to a ring gear (4, 4a, 4b, 4c) via a pinion (5a, 5b, 5c), wherein each ring gear (4, 4a, 4b, 4c) is disposed inside hollow disks (6a, 6b, 6c) stacked on a base such that each disk (6, 6a, 6b, 6c) is one with the ring gear (4, 4a, 4b, 4c), and each disk (6, 6a, 6b, 6c) ) is further combined with a disk head (7, 7a, 7b, 7c) which extends in the same direction as the base and the stack of disks (6, 6a, 6b, 6c), and for each disk head (7, 7a, 7b, 7c) an arm (8, 8a, 8b, 8c) is rotatably connected to the disk head (7, 7a, 7b, 7c) on the one hand and to the platform (2) on the other hand, and each motor (3a, 3b, 3c) is at least partially housed inside at least one disk (6, 6a, 6b, 6c).

Description

[Technical Field]

[0001] The technical field of the invention is actuators for robots, more particularly actuators with three axes of freedom.

[0002] The technical field of the invention is also the control of actuators with three axes of freedom. [Background technology]

[0003] Development of social robots that can have natural interactions with humans in public and domestic spaces, especially for entertainment applications of the robot.

[0004] Such a robot must have the following characteristics in particular: - safe in interactions, especially physical interactions during contact or manipulation carried out by humans, - Allows you to perform dynamic movements to respond to unexpected events as well as to create more enjoyable interactions and cherish the illusion of life. - High-quality animations are executed that allow humans to intuitively understand the robot's intentions, and it is possible to operate in the real physical world by manipulating objects.

[0005] The safety of the operation is due in particular to the mass of the robot or its components, in order to limit the inertia in the control and the risk of injury to the public.

[0006] In robotics, when it is necessary to create an articulated body capable of moving along three axes, the most frequently used technical solution is to chain three actuators in series, as can be seen in most industrial robots and also in humanoid robots at the shoulders and hips.

[0007] The problem with these series articulations is in the mechanics and dynamics.

[0008] From a mechanical standpoint, the ball head's rotational properties, which allow it to rotate in any direction at any time, are only enforced around an initial zone. The further away from this zone you move, the less the ball head's properties are retained. Moving further away, you reach a position known as gimbal lock, where one of the degrees of freedom is lost.

[0009] From a dynamics standpoint, the first actuator in the series supports the mass of the second and third actuators in addition to the useful load. Similarly for the second actuator, which must support the mass of the third actuator. Therefore, to ensure that the articulation can respond well and dynamically, it becomes necessary to oversize the first actuator, which adds strain to the entire system and equally increases its inertia and its cost.

[0010] An alternative to articulated bodies with serial actuators is articulated bodies with parallel actuators. These mechanisms are much more complex to implement and control, but have the advantage that all actuators are fixed to the robot's frame. This makes it possible to both have a much lighter structure in the moving parts and benefit from the application of each of the actuators' power to move the structure. Ultimately, this makes it possible to design highly dynamic systems that can move quickly and precisely.

[0011] The following examples of articulated bodies with parallel actuators are known from the prior art:

[0012] Bulgarelli et al., in Non-Patent Document 1, discloses an articulated body with parallel actuators for the purpose of developing an artificial wrist for sign language telecommunication.

[0013] Non-Patent Document 1 discloses the control of articulated bodies via Euler angle representations. While such representations are conventionally used for such complex systems, they are unstable and can therefore lead to gimbal lock.

[0014] Additionally, as we move away from the initial position of the articulated body, it becomes increasingly difficult to characterize the orientation we are asked to give, since we need to know how to combine rotations about different axes, which becomes complicated in such situations by using the Euler representation.

[0015] Sudki et al., 2003, pp. 131-134, disclose an actuator with three degrees of freedom for marine propulsion, designed to replicate the shoulders of marine animals, particularly penguins.

[0016] Prior art articulated bodies with parallel actuators have the drawback that the frame to which the motors are fixed is of considerable size, making these articulated bodies difficult to integrate into mobile robots. [Prior art documents] [Patent documents]

[0017] [Non-Patent Document 1] A Low-Cost Open Source 3D Printable Dexterous Anthropomorphic Robotic Hand with a Parallel Spherical Joint Wrist for Sign Language Reproduction, Bulgarelli et al., International Journal of Advanced Robotic Systems, Volume 13, Edition 3, January 1, 2016 [Non-patent document 2] Marine Propulsor based on a Three-Degree-of-Freedom Actuated Spherical Joint, Sudki et al., Third International Symposium on Marine Propulsors, smp'13, May 2013 Summary of the Invention [Problem to be solved by the invention]

[0018] The technical problem to be solved is how to benefit from the advantages of parallel actuators by miniaturizing them. [Means for solving the problem]

[0019] One object of the present invention is an articulated body with three degrees of freedom for a robot, comprising a platform and three motors, each connected through a pinion to a ring gear, each ring gear being placed inside hollow disks stacked on a base, each disk being one with the ring gear; Each disk is further associated with its own base and a disk head extending in the same direction as the stack of disks; For each disk head, an arm is rotatably connected to the disk head on the one hand and to the platform on the other hand; Each motor is at least partially housed within at least one disk.

[0020] Each motor is offset relative to the axis of rotation of the disks and can be positioned in a different angular sector to accommodate three motors within the cavity bounded by the base and stacked disks.

[0021] The rest position of each disk head may have an offset angle corresponding to the offset angle between the motors.

[0022] Each motor may further comprise means, in particular a magnetic encoder, for determining the position of the output shaft of the motor relative to a reference position.

[0023] Another object of the invention is to provide a method for controlling an articulated body with three axes of freedom, making it possible to orient a platform according to a required position of a vector linked to the platform and a required angle of rotation about the vector linked to the platform, the method comprising the steps of: determining the coordinates of a three-axis reference frame linked to the platform of the articulating body relative to an initial position and receiving a vector normal to the platform corresponding to a desired position and a desired angle of rotation; determining the coordinates of a rotation vector that allows passing from the initial position of the platform in a three-axis reference frame linked to the platform of the articulating body relative to the initial position to an intermediate position of the platform, the intermediate position being such that passing from the intermediate position of the platform to a required position involves a rotation of the platform itself according to a required rotation angle; determining an initial rotation angle that allows passing from an initial position of the platform to an intermediate position of the platform; determining coordinates of a three-axis reference frame linked to the platform of the articulating body relative to a desired position in the three-axis reference frame linked to the platform of the articulating body relative to an initial position; determining a rotation angle of the first arm according to a desired rotation angle, an initial rotation angle, and coordinates of a three-axis reference frame linked to the platform of the articulated body relative to a desired position in the three-axis reference frame linked to the platform of the articulated body relative to an initial position of a parameter linked to the structure of the articulated body; determining a rotation angle of the second arm according to the desired rotation angle added to the angle offset value, the initial rotation angle, and coordinates of a three-axis reference frame linked to the platform of the articulated body relative to a desired position in the three-axis reference frame linked to the platform of the articulated body relative to an initial position of a parameter linked to the structure of the articulated body; determining a rotation angle of the third arm according to the desired rotation angle subtracted by the angle offset value, the initial rotation angle, and coordinates of the three-axis reference frame linked to the platform of the articulated body relative to a desired position in the three-axis reference frame linked to the platform of the articulated body relative to an initial position of a parameter linked to the structure of the articulated body; Including, For each disk, the rotation angle of the disk is determined according to the rotation angle of the corresponding arm and the coordinates of the three-axis reference frame linked to the platform of the articulating body relative to a desired position in the three-axis reference frame linked to the platform of the articulating body relative to an initial position; Each motor is controlled by the rotation angle of the disk corresponding to the motor.

[0024] The rotation vector can be determined as the vector product between a vector normal to the platform at the initial position and a vector normal to the platform at the desired position.

[0025] The parameters linked to the articular body structures can be the diameter of the distal circle, the diameter of the proximal circle and the center of the proximal circle. The distal circle is the circle traced by the arm around the platform. The proximal circle is the circle traced by the disc.

[0026] The rotation angle can be defined as the arccosine of the scalar product of a vector normal to the platform in its initial position by a vector normal to the platform in its desired position.

[0027] determining a coordinate of a three-axis reference frame linked to the platform of the articulating body relative to a desired position in the three-axis reference frame linked to the platform of the articulating body relative to an initial position, a first quaternion that allows passage from the initial position to an intermediate position, and a second quaternion that allows passage from the intermediate position to the desired position by rotating the platform itself; The desired position of the platform in the reference frame of the platform at its initial position can be determined by successively applying two quaternions to the initial position of the platform.

[0028] The system according to the invention has the advantage that it is reversible, i.e. the articulating body can act as a transducer, in other words the action on the articulating body is converted into an electric current by the motor.

[0029] Other objects, features and advantages of the present invention will become apparent upon reading the following description, given by way of non-limiting example only, and with reference to the accompanying drawings, in which: [Brief explanation of the drawings]

[0030] [Figure 1] FIG. 1 is a three-dimensional view of an articulated body according to the present invention. [Figure 2] FIG. 2 is a side view of an arm according to the present invention. [Figure 3] 1 is a cross-sectional view of a disk according to the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0031] The articulated body according to the invention comprises a system of three concentric shafts that control the actuators, the motors that operate the concentric shafts being integrated inside the articulated body.

[0032] The articulated body 1 allows the platform 2 to be displaced according to three axes of freedom relative to the base by controlling the rotation of three motors.

[0033] The articulated body 1 comprises a base on which are arranged three motors 3a, 3b, and 3c, each connected to a ring gear 4a, 4b, and 4c via a pinion 5a, 5b, and 5c, respectively. In Fig. 1, the motor 3c is shown inside the ring gears 4a, 4b, and 4c. however , motor 3b is disposed inside ring gears 4a, 4b, and motor 3a is disposed inside ring gear 4a. Each ring gear 4a, 4b, 4c is disposed inside hollow disks 6a, 6b, 6c stacked on a base such that each disk 6a, 6b, 6c is mated with the ring gear 4a, 4b, 4c. Each disk 6a, 6b, 6c is further mated with a disk head 7a, 7b, 7c extending in the same direction as the base and the stack of disks 6a, 6b, 6c. Each motor 3a, 3b, 3c is further provided with means for determining the angular position of the motor's output shaft relative to a reference position. Such determining means may be an angular position sensor or a magnetic encoder.

[0034] FIG. 2 shows the connection of the arms with the platform and disk in the general case, which can be displaced for each arm 8a, 8b, 8c.

[0035] For each disk head 7, 7a, 7b, 7c, an arc-shaped arm 8, 8a, 8b, 8c is rotatably connected to the disk head 7, 7a, 7b, 7c on the one hand and to the platform 2 on the other hand. The rotation axis between the disk head 7, 7a, 7b, 7c and the arm 8, 8a, 8b, 8c and the rotation axis between the arm 8, 8a, 8b, 8c and the platform 2 are contained in the same plane as that with the corresponding arm 8, 8a, 8b, 8c. Advantageously, the arc shape represents a quarter of a circle.

[0036] The discs 6, 6a, 6b, 6c form a casing and are provided with bearings that facilitate movement of the discs 6, 6a, 6b, 6c, reduce friction and wear, and make it possible to maintain alignment of the discs 6, 6a, 6b, 6c with respect to the base and between the discs 6, 6a, 6b, 6c.

[0037] In other words, the first disk 6a is connected to the first arm 8a via the disk head 7a, and the first disk 6a is driven by the first motor 3a via the first ring gear 4a and the first pinion 5a.

[0038] The same arrangement is provided for the second disc 6b and the third disc 6c.

[0039] More precisely, the second disk 6b is connected to the second arm 8b via the disk head 7b. 2 The third disk 6c is driven by the second motor 3b through the ring gear 4b and the second pinion 5b. Similarly, the third disk 6c is connected to the third arm 8c through the disk head 7c. 3The third motor 3c drives the second ring gear 4c and the third pinion 5c.

[0040] The first disk 6a is stacked on the second disk 6b, which is itself stacked on the third disk 6c. The third disk 6c is placed on a base. The first arm 6a, the second arm 6b, and the third arm 6c are connected to the platform.

[0041] Referring to FIG. 1, it can be seen that each pinion 5a, 5b, 5c is positioned at a different height from the base to mechanically drive a corresponding ring gear 4a, 4b, 4c. Additionally, to accommodate the three motors 3a, 3b, 3c within the cavity defined by the base and stacked disks 6a, 6b, 6c, each motor 3a, 3b, 3c is offset relative to the axis of rotation of the disks 6a, 6b, 6c and positioned in a different angular sector. This arrangement is shown in FIG. 3. Thus, for the three motors 3a, 3b, 3c, each motor 3a, 3b, 3c is positioned in a different 120° sector.

[0042] This 120° offset angle is also present in the rest position of each disk head 7, 7a, 7b, 7c, with each disk head 7, 7a, 7b, 7c being positioned 120° from the other two.

[0043] Motors 3a, 3b, and 3c are controlled to control the orientation of platform 2, which drives the rotation of each of disks 6a, 6b, and 6c on a circle called the proximal circle, which in turn drives the rotation of mechanically connected arms 8a, 8b, and 8c on another circle called the distal circle.

[0044] 1, it can be seen that there is shown a reference frame R linked to platform 2. The oriented orthonormal three-dimensional reference frame R has an origin O(0,0,0) located at the center of gravity of platform 2, a vector Z(0,0,1) perpendicular to platform 2, a vector Y(0,1,0) extending parallel to the surface of platform 2 and passing through the connection of third arm 8c with platform 2, and a vector Y and Z and vector X(1,0,0), which are orthogonal to vector Y and form a oriented orthonormal frame of reference. Recall that a 3D oriented frame of reference is a reference frame, the angle between vector X and vector Y is the directional angle, the angle between vector Y and vector Z is the directional angle, and the angle between vector Z and vector X is the directional angle.

[0045] We now turn our attention to the control of the system that allows the platform 2 to be positioned in place.

[0046] To prevent gimbal lock, a representation in the form of a quaternion is used: a quaternion is a normalized vector in three-dimensional space combined with a rotation around this vector by an angle θ.

[0047] The following dimensions are defined: R0 = (X0, Y0, Z0): Oriented orthogonal reference frame defining the initial position of platform 2 Rreq = (Xreq, Yreq, Zreq): Oriented orthogonal reference frame defining the desired position of platform 2 V = (a, b, c): rotation vector perpendicular to the plane defined by vectors R0 and Rreq θ: rotation angle around vector V that defines the desired position for platform 2 β: rotation angle of the platform 2 itself between the initial position and the desired position

[0048] The platform 2 is required to be controlled to pass from an initial position defined by the reference frame R0 to a required position defined by the reference frame Rreq.

[0049] The motion is then decomposed into two rotations: The first is a rotation by an angle θ about a vector V normal to the plane defined by the reference frames R0 and Rreq.

[0050] The second rotation is a rotation by β about the vector normal to platform 2. Note that the normal vector to platform 2 can be vector Z0, Zreq, or any other intermediate vector.

[0051] The order of rotations is irrelevant, so the second rotation can be performed before the first.

[0052] The vector V is then defined as the vector product of the vectors Z0 and Zreq in the following way: [Formula 1]

number

[0053] The rotation angle θ is defined according to the components of R0 and Rreq that are perpendicular to the platform 2. [Formula 2]

number

[0054] The vector V is then normalized so that it can be used with the quaternion.

[0055] The first quaternion q1 associated with the first rotation by angle θ about vector V is written as follows: [Formula 3]

number

number

number

number

number

[0056] A second quaternion q2 is then defined that makes it possible to perform a second rotation by an angle β around a vector perpendicular to the platform 2. By using the same format as presented for the first quaternion q1, we obtain: [Formula 8]

number

number

number

number

number

[0057] The vector perpendicular to platform 2 is a vector of the form (0,0,Z), where the quaternion terms x2 and y2 are 0.

[0058] There are therefore two quaternions associated with two rotations that allow passing from the reference frame R0 linked to the initial position to the reference frame Rreq linked to the desired position.

[0059] However, it is still necessary to determine the rotation angles of the three motors, noted θ11, θ12, and θ13, in order for the arm to perform the rotations defined by these quaternions to move the platform 2 from the initial position R0 to the required position Rreq.

[0060] To determine these rotation angles, the following factors are defined: Rd: Radius of the circle traced by the arm around platform 2, also called the distal circle Rp: Radius of the circle traced by the disc, also called the proximal circle Cp=(0,0,Cz): coordinates of the center of the circle traced by the disc, also called the proximal circle Pc=(0,0,0): Coordinates of the center of the distal circle θ3i Angle of arm i relative to platform 2 θ1i angle of disk i relative to its reference position

[0061] Distal circle Xi and proximal circle Xi p The formula is expressed by the following formula: [Formula 13]

number

number

[0062] Thus, by applying the first quaternion q1, a first displacement is defined that allows passing from an initial position defined by the reference frame R0 to an intermediate position defined by the reference frame Rinter. [Formula 15]

number

[0063] Thus, by applying the second quaternion q2, a second displacement is defined that allows passing from the intermediate position defined by the reference frame Rinter to the desired position defined by the reference frame Rreq. [Formula 16]

number

number

number

[0064] However, the rotation defined by the second quaternion q2 is performed around the direction Z. The direction Z is therefore invariant, from which it follows that Zinter = Zreq.

[0065] If the rotation vector is normalized, the conjugate of the quaternion q1

number

number

number

[0066] Since there is no rotation about the axes X and Y during the application of the second quaternion q2, the calculation of the coordinates Xreq and Yreq can be performed in a single step involving the two quaternions q1 and q2. [Formula 19]

number

number

[0067] With changes in the reference frame, the coordinates X0, Y0, Z0 and the coordinates Xreq, Yreq, Zreq are defined in the reference frame R0 linked to the initial position of the platform.

[0068] Therefore, in reference frame R0, we have: X0=(X0,0,0) Y0=(0,Y0,0) Z0=(0,0,Z0) Xreq=(Xx,Yx,Zx) Yreq=(Xy,Yy,Zy) Zreq=(Xz,Yz,Zz)

[0069] Knowing the coordinates Xreq, Yreq, Zreq, it is possible to determine the angles θ11, θ12, θ13 of the disc. By solving the equation Xi=Xp that arises from equations 1 and 2, the following simultaneous equations are obtained: [Formula 21]

number

number

number

[0070] By solving these simultaneous equations [Formula 21], [Formula 22], and [Formula 23], the following equations for angles θ3i and θ1i can be obtained, respectively. [Formula 24]

number

number

[0071] By applying equation [Equation 24], the angle θ3i of each arm i in the system is first determined. By applying equation [Equation 25], the angle θ1i of each disk i in the system is then determined.

[0072] As mentioned above, the vector Zreq is not modified by the rotation by the second rotation angle β, whereas the vectors Xreq and Yreq are certainly modified by this rotation.

[0073] When stationary, the disks are offset by 120° relative to each other. Using the angular position of the first disk as a reference, it is possible to obtain the angular position of the second disk by applying a +120° offset to the second rotation angle β, and the angular position of the third disk by applying a -120° offset to the second rotation angle β.

[0074] It follows from this that the values ​​Xx, Yx and Zx in equations [Equation 24] and [Equation 25] are modified due to this offset.

[0075] The embodiment described above includes a second rotation along axis Z of the reference frame linked to the platform. In other embodiments, the second rotation can be performed along axis X or axis Y, or along a vector linked to the platform. Those skilled in the art will be able to determine the above rotation according to the axis considered. description The mathematical formula will be adapted accordingly.

[0076] Likewise, the first and second rotations may be reversed and interchanged without departing from the scope of the present invention.

[0077] The method for controlling an articulated body with three axes of freedom that allows orienting the platform 2 according to a required position and a required angle of rotation of the platform 2 comprises the following steps:

[0078] During the first step, the coordinates of the oriented orthonormal three-axis reference frame linked to the articulated platform 2 for an initial position are determined, and the coordinates of the oriented orthonormal three-axis reference frame linked to the articulated platform 2 for a required position and a required rotation angle are determined.

[0079] During a second step, the coordinates of a rotation vector V are determined that allow passage from a reference frame linked to the initial position of the platform 2 to a reference frame linked to an intermediate position of the platform 2 in the three-axis reference frame linked to the platform 2 of the articulating body relative to the initial position, the intermediate position being such that passage of the platform 2 from the intermediate position to the required position involves rotation of the platform 2 itself according to the required rotation angle.

[0080] During the third step, an initial rotation angle is determined that allows the platform 2 to pass from its initial position to its intermediate position.

[0081] During a fourth step, the coordinates of the three-axis reference frame linked to the articulating platform 2 relative to the required position in the three-axis reference frame linked to the articulating platform 2 relative to the initial position are determined.

[0082] During a fifth step, the rotation angle θ31 of the first arm is determined according to the coordinates of the three-axis reference frame linked to the platform 2 of the articulated body relative to the required position in the three-axis reference frame linked to the platform 2 of the articulated body relative to the required rotation angle, the initial rotation angle and the initial position of the parameters linked to the structure of the articulated body.

[0083] During a sixth step, the rotation angle θ32 of the second arm is determined according to the desired rotation angle added to the angle offset value, the initial rotation angle and the coordinates of the three-axis reference frame linked to the articulated platform 2 relative to the desired position in the three-axis reference frame linked to the articulated platform 2 relative to the initial position of the parameters linked to the articulated structure.

[0084] During a seventh step, the rotation angle θ33 of the third arm is determined according to the required rotation angle minus the angle offset value, the initial rotation angle and the coordinates of the three-axis reference frame linked to the articulated body's platform 2 relative to the required position in the three-axis reference frame linked to the articulated body's platform 2 relative to the initial position of the parameters linked to the articulated body's structure.

[0085] During an eighth step, for each disk the rotation angle of the disk is determined according to the rotation angle of the corresponding arm and the coordinates of the three-axis reference frame linked to the articulating platform 2 relative to the required position in the three-axis reference frame linked to the articulating platform 2 relative to the initial position.

[0086] Each motor is controlled by the rotation angle of the disk corresponding to the motor. [Explanation of symbols]

[0087] 1 articulated body, 2 platform, 3a first motor, 3b second motor, 3c third motor, 4a first ring gear, 4b second ring gear, 4c third ring gear, 5a first pinion, 5b second pinion, 5c third pinion, 6 disk, 6a first disk, hollow disk, 6b second disk, hollow disk, 6c third disk, hollow disk, 7 disk head, 7a first disk head, 7b second disk head, 7c third disk head, 8 arm, 8a first arm, 8b second arm, 8c third arm

Claims

1. A method for controlling an articulated body having three degrees of freedom for a robot, comprising: The articulating body is Platform (2), Three motors (3a, 3b, 3c) each connected to a ring gear (4, 4a, 4b, 4c) via a pinion (5a, 5b, 5c); Equipped with each of said ring gears (4, 4a, 4b, 4c) is placed inside a hollow disc (6a, 6b, 6c) stacked on a base, so that each of said discs (6, 6a, 6b, 6c) is one with the ring gear (4, 4a, 4b, 4c); each of said disks (6, 6a, 6b, 6c) is further united with a disk head (7, 7a, 7b, 7c) which itself extends in the same direction as said base and the stack of said disks (6, 6a, 6b, 6c); for each said disk head (7, 7a, 7b, 7c) an arm (8, 8a, 8b, 8c) is rotatably connected to said disk head (7, 7a, 7b, 7c) on the one hand and to said platform (2) on the other hand, each of said motors (3a, 3b, 3c) is at least partially housed inside at least one of said disks (6, 6a, 6b, 6c); It is an articulated body, The method comprises:

1. A method making it possible to orient a platform (2) according to a required position of a vector linked to said platform (2) and a required angle of rotation about said vector linked to said platform (2), comprising: - determining the coordinates of a three-axis reference frame linked to the platform (2) of the articulated body relative to an initial position and receiving the vector normal to the platform (2) corresponding to the required position and required angle of rotation; - determining the coordinates of a rotation vector allowing passage from the initial position of the platform (2) in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position to an intermediate position of the platform (2), the intermediate position being such that the passage of the platform (2) from the intermediate position to the required position involves a rotation of the platform (2) itself according to a required rotation angle; - determining an initial rotation angle that allows passing from said initial position of said platform (2) to an intermediate position of said platform (2); - determining the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the desired position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position; - determining a rotation angle of the first arm according to the required rotation angle, the initial rotation angle and the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the required position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position of a parameter linked to the structure of the articulated body; - determining the rotation angle of the second arm according to the required rotation angle added to an angle offset value, the initial rotation angle and the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the required position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position of a parameter linked to the structure of the articulated body; - determining the rotation angle of the third arm according to the required rotation angle subtracted by the angular offset value, the initial rotation angle and the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the required position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position of a parameter linked to the structure of the articulated body; Including, For each disk, the rotation angle of the disk is determined according to the rotation angle of the corresponding arm and the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the required position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position, A method wherein each of the motors is controlled by the rotation angle of the disk corresponding to the motor.

2. 2. The method of claim 1, wherein the rotation vector is determined as a vector product between the vector perpendicular to the platform (2) in the initial position and the vector perpendicular to the platform (2) in the required position.

3. The method of claim 1 , wherein the parameters linked to the structure of the articular body are a diameter of a distal circle, a diameter of a proximal circle, and a center of the proximal circle.

4. 2. The method of claim 1, wherein the rotation angle is defined as the arccosine of the scalar product of a vector normal to the platform in the initial position by a vector normal to the platform in the desired position.

5. In order to determine the coordinates of the three-axis reference frame linked to the platform (2) of the articulated body relative to the desired position in the three-axis reference frame linked to the platform (2) of the articulated body relative to the initial position, a first quaternion that allows passage from the initial position to the intermediate position and a second quaternion that allows passage from the intermediate position to the desired position by rotating the platform itself are determined; applying two of the quaternions successively to the initial position of the platform to determine the desired position of the platform in the reference frame with the platform in the initial position. The method of claim 1 further comprising:

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