NEW ROBOTIC JOINT CONFIGURATION

Technical Field

The invention is a new mechanism (mechanical configuration) that simulates a ball-socket joint with its three degrees of freedom, where this mechanism is installed between two mechanical links, it has advantages over the currently employed mechanisms, as it doesn't suffer from several drawbacks they do suffer from.

Background Art

To simulate a ball-joint (with three degrees of freedom, about three perpendicular axes) between mechanical links Ll and L2, a mechanism K is installed between them. Since there are only three degrees of freedom (Dl, D2, and D3) no more than three inputs should be allowed to operate this mechanism. In case more than three inputs are applied, the mechanism is considered redundant and excessively complicated. The mechanisms currently employed are usually one of two designs.

Mechanism Kl (shown in Fig.l)

• The axis ofDl is perpendicular to the axis of D2, and is aligned to the axis of L 1.

• The axis of D3 is perpendicular to the axis of D2, and is aligned to the axis of L2.

• The three axes of Dl, D2, and D3 intersect at a common point (C) which is also the center of rotation of the simulated ball-joint.

• The axis of Dl is spatially fixed with respect to the axis of Ll and KIa.

• The axis of D2 is spatially fixed with respect to the axis of Kl a and Kl b.

• The axis of D3 is spatially fixed with respect to the axis of L2 and K2b.

Mechanism K2 (shown in Fig.2)

(Also known as Universal Joint, or Hook's Joint)

• The axis of Dl is perpendicular to the axis of D2 and Ll, and intersects them at C.

• The axis of D3 is perpendicular to the axis of D2, and is aligned to the axis of L2.

• The three axes of Dl, D2, and D3 intersect at a common point (C) which is also the center of rotation of the simulated ball-joint.

• The axis of Dl is spatially fixed with respect to the axis of Ll or K2a, and K2b.

• The axis of D2 is spatially fixed with respect to the axis of K2b, and K2c.

• The axis of D3 is spatially fixed with respect to the axis of L2, and K2c. Terminology

Twist angle

In the initial position of a ball-joint shown in Fig.3a, each of the links surfaces is divided into eight equal longitudinal segments each is represented by a numbers and a degree of shading. Each segment on Ll has a corresponding segment on L2 with the same number and shading. (These numbered segments are for demonstration purposes only). The following cases help to form some sense for the definition of the twist angle. a) Axes of Ll and L2 are aligned (Fig.3bl and Fig.3b2). b) Axes of Ll and L2 are not aligned (Fig.3cl and Fig.3c2). As shown in Fig.3d:

• Plane P is formed by the axes of Ll and L2 (where the deviation angle is measured).

• The deviation angle is measured from the axis of Ll to the axis of L2.

• Plane P intersects link Ll in plane Pl

• Plane P2 on link L2 corresponds to Pl on Ll in the initial position Note: hi all figures, θ is the deviation angle, and τ is the twist angle.

Definition of twist angle

It is the angle measured from plane P to plane P2, and is measured counter clockwise about the vector passing through the axis of link L2 protruding from the center point to the mechanical link L2.

From the previous definition, note that the twist angle in Fig.3d is negative. Note: If the axes of Ll and L2 are aligned, an infinite number of planes may be used to measure the twist angle. Otherwise only one plane (P) could be used to measure the twist angle.

Drawbacks of mechanism 1 1. Locking Delay

If axes of Dl and D3 are aligned, a rotation about an axis perpendicular -only at this instant- to the axes of Dl and D2 is not possible, unless Dl rotates an angle of 90° until the axis of D2 is aligned to the desired axis (where Ll is fixed, and L2 rotates), in other words the mechanism in a certain position is locked to rotate about a certain axis which causes a delay. Example:

In Fig.4a, L2 is required to move from point Z to point X, then to point Y on the path shown in the figure. After reaching point X (as shown in Fig.4b) it is impossible to reach point Y without rotating Dl 90°. D3 will rotate 90° in the opposite rotation direction of Dl to maintain a zero twist angle between the links Ll and L2 (as shown in

Fig.4c). Finally, D2 will rotate until L2 reaches point Y (as shown in Fig.4d). Notes:

• Path ZX is a quarter-circular planar path.

• Path XY is a quarter-circular planar path.

• Path ZX and XY lie in two plans perpendicular to one another. 2. Undesired Twist

Link L2 is twisted relative to Ll as a result of rotating Dl from its initial position without rotating D3 (where Ll is fixed and L2 rotates). This mechanism needs only to rotate Dl and D2 to change the position of L2 in space (as shown in Fig.5a). D3 is used to rotate L2 about its axis to achieve the required twist. The drawback is that the purpose of D3 initially was to control the twist angle, which still occurred without rotating D3.

In order to rotate L2 in space without any twist between L2 and Ll, Dl, D2, and D3 must all be rotated at the same time. In addition, Dl and D3 must rotate at the same synchronized speed as shown in Fig.5b. Drawbacks of mechanism 2

1. Undesired Twist

Link L2 is twisted relative to Ll as a result of rotating Dl and D2 simultaneously from their initial positions without rotating D3 (where Ll is fixed and L2 rotates). This mechanism needs only to rotate Dl and D2 to change the position of L2 in space (as shown in Fig.όa). D3 is used to rotate L2 about its axis to achieve the required twist. The drawback is that the purpose of D3 initially was to control the twist angle, which still occurred without rotating D3. Note'. The angle values in Fig.όa and Fig.όb are approximated to the nearest integer.

In order to rotate L2 in space without any twist between L2 and Ll, Dl, D2, and D3 must all be rotated at the same time. In addition, Dl, D2, and D3 must all rotate in synchronization with each other in a complex relationship as shown in Fig.όb.

2. Movement constrain (for high deviation angle)

As shown in Fig.7a, L2 cannot rotate about the axis of Ll while maintaining a constant deviation angle of 90° or more (where Ll is fixed and L2 rotates). This is due to the collision between M2a and M2c (collision zones are pointed out with arrows in the figure). However, Ll can rotate easily about the axis of L2, if L2 is fixed. D3 rotates, while

Dl and D2 do not. Even so, a twist angle will always exist between Ll and L2 which cannot be completely and permanently eliminated.

Disclosure of the Invention

• The invention is a mechanism which simulates a ball-joint (with three degrees of freedom about three perpendicular axes) between two mechanical links, Ll and L2.

• In this mechanism, degrees of freedom Dl (or Dl*) and D2 are responsible for rotating L2 relative to Ll in space without making any twist angle between the two links.

• D3 alone controls the twist angle between Ll and L2.

• This mechanism does not suffer from any of the drawbacks found in Mechanisms 1 and 2 (as mentioned in the Background Art), which are: o Undesired twist without rotating D3. o Locking delay regardless of the relative positions of Ll and L2. o Movement constrain (for large deviation angle)

• The deviation angle between Ll and L2 can reach 120° with ease, and any of the links can rotate about the axis of the other while maintaining a constant deviation angle without causing any twist.

1- Configuration I As shown in Fig.8a:

• Either of the pyramid-shaped component Rl or the conical-shaped component Ol is fixed to the link Ll .

• A rotational degree of freedom (Dl-I) exists between the first arc Al-I and the component Pl about a spatially fixed axis relative to Rl .

• A rotational degree of freedom (D2-1) exists between the second arc A2-1 and the component Rl about a spatially fixed axis relative to Rl, whereas this axis is perpendicular to that of Dl-I and intersects it at a point C, which is the center of the simulated ball-joint.

• Either of the pyramid-shaped component R2 or the conical-shaped component 02 is fixed to the link L2*.

• A rotational degree of freedom (D 1-2) exists between the first arc A 1-2 and the component R2 about a spatially fixed axis relative to R2. Another rotational degree of freedom (H2) exists between arcs Al-I and Al -2. The axis of H2 passes through the center point C. • A rotational degree of freedom (D2-2) exists between the second arc A2-2 and the component R2 about a spatially fixed axis relative to R2. This axis is perpendicular to that of Dl-2 and intersects it at point C. Another rotational degree of freedom (Hl) exists between the arc A2-1 and the arc A2-2. The axis of Hl passes through the center point C. Notes:

• The axis of Ll as specified is perpendicular to Dl-I and D2-1 and intersects them at the center point C. It is also spatially fixed relative to Rl .

• The axis of L2* as specified is perpendicular to Dl-2 and D2-2 and intersects them at the center point C. It is also spatially fixed relative to R2.

• The vertex of each pyramid-shaped component is at point C. The pyramid shaped components have 4 edges, where each two opposite edges form a plane. This plane is perpendicular to the axes of rotation of one of the arc-shaped components connected to its respective pyramid-shaped component.

• Rl may be replaced with Ol (as shown in Fig.8b).

• R2 may be replaced with 02 (as shown in Fig.8b).

• The vertex of the conical-shaped components is at point C. The axis of the cone is aligned with the axis of its respective mechanical link.

1.1- Theory of operation

• Both degrees of freedom Hl and H2 are driven and rotate as a result of the rotation of degrees of freedom Dl and D2 as will be demonstrated.

• The axis of Hl and H2 form a plane M. About which, Rl is symmetric to R2, Al-I to Al-2, A2-1 to A2-2, and ultimately Ll to L2*. This will be referred to as the mirror rule later on in this document.

• As Dl-I rotates, Dl-2 also rotates satisfying the mirror rule. This is a motion constraint between Dl-I and Dl-2. This forms the degree of freedom Dl where its axis is aligned with the axis of the hinge Hl .

• As D2-1 rotates, D2-2 also rotates satisfying the mirror rule. This is a motion constraint between D2-1 and D2-2. This forms the degree of freedom D2 where its axis is aligned with the axis of the hinge H2.

• As a result of this symmetric motion, no twist is possible between Rl and R2, or Ll and L2*. • D3 is the degree of freedom existing between L2* and L2, and it controls the angle of twist between Ll and L2.

• Link L2 has three degrees of freedom relative to Ll. The two degrees of freedom Dl and D2 which exist between Ll and L2* are a major subject of this patent.

• The angle between the axes of Dl and D2 is variable, while the angle between the axes of rotation of two arc-shaped components on the same pyramid-shaped component is always a right angle.

• If Dl-I is fixed, then the axis of D2 is spatially fixed relative to any of the pyramid- shaped components, even if D2-1 is rotating.

• If D2-1 is fixed, then the axis of Dl is spatially fixed relative to any of the pyramid- shaped components, even if Dl-I is rotating.

1.2- Mathematical Equations As shown in Fig.9a:

• The coordinate system used in this section is similar to the spherical coordinate system. Where, the angle φ in plane N is formed between the projection of the axis of L2* on N and Vector -V, the angle φ is measured in the counter clockwise direction around the vector that is aligned with the axis of Ll and is directed from Ll to the center point (C) of the mechanism.

• Plane N is perpendicular to the axis of Ll and intersects the link Ll anywhere along Ll.

• Vector -V exists in Plane N, intersects with the axis of Ll, and is parallel to the axis ofD2-l.

• Vector U exists in Plane N, intersects with the axis of Ll, and is parallel to the axis of Dl-I.

• The projection of L2* is also the intersection of planes N and P. Note that the axes of Ll and L2* lie in plane P. The deflection angle is measured on plane P, as mentioned in the definition of the twist angle earlier in this document.

• Angle θ exists in plane P, and is measured from the axis of Ll to the axis of L2* . As shown in Fig.9b:

• The mechanism is controlled using Dl-I and D2- 1 , where Dl-I rotates an angle of β _s , and D2-1 rotates an angle of ζ _s . Therefore, a mathematical relation will be derived between β _s & ζ _s , and φ & θ. • As Dl-I and D2-1 rotate angles of β _s & ζ _s respectively, mirror plane M rotates, forming an angle of θ _s with plane N.

• Since plane M is the mirror plane, it is perpendicular to plane P. In addition, plane P is perpendicular to both plane M and plane N. Therefore, angle θ _s exists in plane P.

• Since plane M is the mirror plane, angle θ _s is equal to half the angle θ. This is due to the fact that the angle that lies between any two planes is equal to the angle formed by the two normals to the planes.

From the geometry shown in Fig.9b:

• Taking the distance between the center point C and plane N to be λ. The value of λ is variable due to the fact that the position of plane N along the axis of Ll is insignificant. Regardless of the value of λ, the relations between all the angles will remain intact. The value of λ will be used to derive the mathematical equations, upon derivation the variable λ will be cancelled out automatically. This is the same for the distances γ and α shown in Fig.9b

^■ As mentioned

From the geometry

From the triangle λ, -V, D2

From the triangle λ, U, Dl

From the triangle α, U, -V, and applying Pythagoras Substituting eqn.0.2 and eqn.0.3 in the previous equation

From the triangle α, U, -V, and the concept of triangles similarity

Substituting eqn.0.2 and eqn.0.3 in the previous equation

Substituting eqn.0.5 in eqn.0.1 Substituting eqn.0.4 in the previous equation

Substituting eqn.l in the previous equation and simplifying

■ From the triangle α, U, -V Substituting eqn.0.2 and eqn.0.3 in the previous equation

■ Form the triangle α, U, -V Substituting eqn.0.1 and eqn.0.2 in the previous equation

And since sin(^) = tan(^) • cos(^) Substituting eqn.2 and eqn.3 in the previous equation

■ Differential equations

Squaring eqn.2, then differentiating, and simplifying

Differentiating eqn.3 and simplifying

Differentiating eqn.4 and simplifying

Differentiating eqn.4.1 and simplifying q

Summary:

• Angles β _s and ζ _s have a theoretical range of [-90, +90] degrees, practically a range of [-60, +60] degrees is sufficient.

• Angle β _s is measured in the counter clockwise direction about vector U (illustrated in Fig.9b). Note that vector U is parallel to Dl-I as mentioned before.

• Angle ζ _s is measured in the counter clockwise direction about vector V (illustrated in Fig.9b). Note that vector V is parallel to D2-1 as mentioned before.

• In the initial position of the mechanism, Ll and L2* are aligned. Therefore, they are perpendicular to both Dl and D2. In addition, Dl and D2 are perpendicular to each other only at this instant.

• In the initial position, the angles β _s , ζ _s , θ _s , φ, and θ are equal to zero.

• Note that the relations between β _s & ζ _s and φ & θ are trigonometric relations.

• The range of angle φ is [0, 360] degrees. Therefore, it may exist in any of all 4 quadrants, in order to calculate its value, two trigonometric relations are required (eqn.3 and eqn.4).

• The theoretical range of angle θ is [0, 180] degrees but practically, a range of [0, 120] degrees is sufficient. Therefore it may exist in the 1 ^st or 2 ^nd quadrant, and in order to calculate its value, a single trigonometric relation is required (eqn.2).

• The input to this system is β _s and ζ _s , and the output is φ and θ.

• Eqn.5 shows the variation of the speed of θ with the speed of β _s and ζ _s .

• Eqn.6 shows the variation of the speed of φ with the speed of β _s and ζ _s .

• Eqn.7 shows the required speed of β _s to obtain certain desired values for the speeds of φ and θ.

• Eqn.8 shows the required speed of ζ _s to obtain certain desired value for the speeds ofφ and θ. • Note that in the differential equations, the absolute values of the angles β _s , ζ _s , φ, θ _s , and θ exist alongside the rate of change of these angles. Therefore, the rate of change of any angle at any certain instant depends on the absolute values of the angles at that same instant. 1.3- Means of actuation o It is mentioned in the theory of operation that, as Dl-I rotates, D 1-2 rotates satisfying the mirror rule, and as D2-1 rotates, D2-2 rotates satisfying the mirror rule. o But it wasn't mentioned how this symmetry happens, how this constrains between an arc-shaped component and its corresponding arc-shaped component happens, well there are many ways to do this.

• Transmitting the motion to the opposite side using strings As shown in Fig.lla

^■ The strings are connected on the paths shown in the figure.

^■ The mechanism is set to the initial position, and all the strings are tensioned and not lose.

^■ Each side of string Sl is fixed with its bolt on the component Rl as shown.

^■ Each side of string S2 is fixed with its bolt on the component R2 as shown.

^■ Considering Rl to be fixed, if string S 1 is pulled from the right bottom side, arc Al-I will rotate in clockwise direction, with an absolute speed of dβj dt to get closer to Rl, the axis of H2 exists on arc- shaped component

Al-I, string S2 is pulled with the string Sl which makes the component R2 rotate in the same direction, but with an absolute speed equal to double the speed dβ _s /dt, and that's because string Sl is folded on itself, that makes the speed of the component R2 relative to the axis of H2 equal dβ _s I dt, that means that both Rl and R2 are symmetric around H2 which forms mirror plane M, and that's how we get the symmetry in degree of freedom Dl

^■ The blue string passes through a hole, that hole exists on a pin where its axis coincides with the axis of H2, so even if D2 is rotated, and there is an angle between Al-I and Al -2, the strings length will not change on its paths and will not be affected.

^■ The amount of string pulled from a side is supplied to the opposite side, which makes the string always tensioned.

^■ Of course degree of freedom D2 works the same way as explained with D 1. ^■ The amount of string pulled is linearly proportional to the angle β _s , this is also the case with the string responsible for the angle ζ _s . This linear proportionality is a great advantage.

^■ The flaw in this method is the friction due to the slipping between the strings on each other, and also on the arc-shaped components.

• Transmitting the motion to the opposite side using hydraulic fluid As shown in Fig.llb

^■ If two hydraulic cylinders are connected as shown, their motion become similar and constrained to each other, if one of them advances, the hydraulic fluid pushes the other to advance as well, and vise versa, note that the hydraulic fluids are named Fl and F2 just for explanation.

As shown in Fig.llc

^■ As shown if a hydraulic cylinder is fixed on each of the pyramid-shaped components, and the motion of the strings is constrained to the hydraulic cylinders, therefore if an arc-shaped component rotate relative to its respective pyramid-shaped component, the motion is transferred to the hydraulic cylinder, consequently to the opposite hydraulic cylinder, that makes the corresponding arc-shaped component rotates relative to its pyramid-shaped component, and the required symmetry accrues satisfying the mirror rule.

^■ The strings are connected as shown.

^■ Each end of string S3 is connected to its pin on the hinge H2 as shown.

^■ Each end of string S4 is connected to its pin on the hinge H2 as shown.

^■ The mechanism is set to the initial position, and all strings are tensioned and not lose.

^■ The amount of string pulled is linearly proportional to the angle β _s , and this is also the case with the string responsible for the angle ζ _s .

■ Transferring the motion to the other side using fluid is more efficient, while the strings in this case cause no slipping and friction like the previous way of actuation.

• Other methods of actuation

^■ As shown in Fig.12a which operates completely with hydraulic fluid, where the two fluids F3 and F4 are responsible for moving the arc-shaped component, and the two fluids F5 and F6 are responsible for satisfying the required symmetry to the opposite arc-shaped component, the valve shown when set on the most right position the mechanism becomes passive, when setting it to 2 ^nd position from the right, the mechanism rotates in counter clockwise direction, when set to the 3 ^rd position from the right, the mechanism rotates in the clockwise direction, when setting the valve on the last position, the mechanism is brake locked, of course the two other arc- shaped components are set with a similar actuation method. 1.4- Pyramid-shaped components and conical-shaped components:

• If pyramid-shaped component Rl is fixed on link Ll, and Pyramid-shaped component R2 on link L2*, both angles β _s , and ζ _s can reach their maximum value simultaneously, that means that any of these two angles can change on all its range independent of the value of the other angle, therefore the pyramid-shaped components increase the workspace of the mechanism, that makes angle θ reach more than 120 degrees on most of the range of angle φ, but angle θ cannot exceed 120 degrees at four specific values for angle φ, these four values are [0, 90, 180, 270] degrees, when both angles β _s , and ζ _s reach their max value which is 60 degrees, angle θ equals approximately 135.58 degrees.

• However, if conical-shaped component Ol is fixed on link Ll, and the conical- shaped component 02 on link L2*, angles β _s , and ζ _s can't reach their max value simultaneously, any angel of these two cannot change on all of its range, unless the other one equals zero, here the workspace is enclosed by a conical surface, that means that the maximum value of angle θ is always 120 degrees.

• These pyramid-shaped, conical-shaped or whatever shaped components are the end limits to the motion of the mechanism, when one touches the opposite. They could be shaped with any profile to get a specifically desired workspace, it is not a must that the part fixed on Ll is similar to the one fixed on L2*.

2- Configuration II As shown in Fig.8c: Configuration II is very similar to Configuration I, with few differences:

• Arc-shaped component A2-1 doesn't exist and its degree of freedom (D2-1).

• Arc-shaped component A2-2 doesn't exist and its degree of freedom (D2-2), and there is no hinge Hl. • The vertex of the pyramid-shaped components is point C, the pyramid-shaped components have 4 edges, and each two opposite edges form a plane, where one of these two planes is perpendicular to the axis of the arc-shaped component connected to its respective pyramid-shaped component. Other form of configuration II (as shown in Fig.δd)

• More complex, but necessary to clear some points.

• In each pyramid-shaped component the axis of an arc-shaped component is spatially fixed relative to this pyramid-shaped component, this axis is also perpendicular to the axis of the mechanical link fixed on this pyramid-shaped component, and interests it at point C. The axis of the other arc-shaped component is also perpendicular to the axis of the mechanical link fixed on this pyramid-shaped component and interests it at point C, but it is not spatially fixed relative to this pyramid shaped component, and not necessarily perpendicular to the axis of the previously mentioned arc-shaped component.

• Each part mentioned in the previous point has its similar corresponding part, which is symmetric to it around plane M.

• Each arc-shaped component on a side is hinged to its corresponding arc-shaped component on the other side, ultimately there is two hinges.

• The axes of these two hinges pass through point C ₅ and are connected with a part that makes them perpendicular to each other.

2.1- Theory of operation It is very similar to the theory of operation of configuration I.

• The hinge H2 is not driven this time, and is directly controlled to become the degree of freedom D2.

• Mirror plane (M) contains the axis of D2. About that plane, Rl is symmetric to R2, Al-I to Al-2, and ultimately Ll and L2*. This is known as the mirror rule.

• As Dl-I rotates, D 1-2 also rotates satisfying the mirror rule. This is a motion constraint between Dl-I and Dl-2. This forms the degree of freedom Dl*, the axis of Dl * is perpendicular to the axis of D2, and lies hi the mirror plane M.

• As a result of this symmetric motion, no twist is possible between Rl and R2, or Ll and L2*.

• D3 is the degree of freedom existing between L2* and L2, and it controls the angle of twist between Ll and L2. • Link L2 has three degrees of freedom relative to Ll. The two degrees of freedom Dl* and D2 which exist between Ll and L2* are a major subject of this patent.

• The axes of Dl* and D2 are always perpendicular to each other, while (in the other form of configuration II) the angle between the axes of rotation of two arc-shaped components on the same pyramid-shaped component is variable.

• If Dl-I is fixed, then the axis of D2 is spatially fixed relative to any of the pyramid- shaped components, even if D2 is rotating.

• If D2 is fixed while Dl-I is rotating, then the axis of Dl* is not spatially fixed relative to any of the pyramid-shaped components.

2.2- Mathematical Equations

• The mechanism is controlled using Dl-I and D2, where Dl-I rotates an angle of β _s , and D2 rotates an angle of ψ, therefore, a mathematical relation will be derived between β _s & ψ , and φ & θ.

• Angle ψ is the angle which arc D 1 -2 rotates relative to arc D 1 - 1.

• Since plane M is a mirror plane, it rotates an angle of ψ _s relative to arc Al-I, where ψ _s is equal to half the angle ψ.

Review the geometry shown in Fig.9b:

^■ From the geometry

^■ It is mentioned before that Substituting eqn.9 and eqn.10 in eqn.2

Substituting eqn.9 and eqn.10 in eqn.3

Substituting eqn.l 1 in eqn.4, and simplifying ^■ Differential equations

Squaring eqn.ll, then differentiating and simplifying

... eqn.14

Differentiating eqn.12 and then simplifying

... eqn.15

Remember eqn.7 eqn.7 _

Squaring eqn.l 1, substituting using eqn.12 and simplifying Differentiating the previous equation and then simplifying

Summary:

• Eqn.9 helps to transform from the variables of configuration I to the variables of configuration II.

• Angle ψ has a theoretical range of [-180, +180] degrees, practically [-120, +120] degrees is a sufficient range.

• Angle ψ is measured in the counter clockwise direction about the vector of D2, where vector V is the projection of the vector of D2.

• In the initial position of the mechanism, Ll and L2* are aligned, therefore they are perpendicular on both Dl* and D2.

• In the initial position the angles β _s , ψ, ψ _s , θ _s , φ, and θ equal to zero.

• Note that the relations between β _s & ψ and φ & θ are trigonometric relations.

• Angle φ as mentioned before requires two trigonometric relationships to calculate it Ceqn.12 and eqn.13).

• Angle θ as mentioned before requires only one trigonometric relationship to calculate it Ceqn.l 1). • The input to this system is β _s and ψ, and the output is φ and θ.

• Eqn.14 shows the variation of the speed of θ with the speed of β _s and ψ.

• Eqn.15 shows the variation of the speed of φ with the speed of β _s and ψ.

• Eqn.7 shows the required speed of β _s to obtain certain desired values for the speeds ofφ and θ.

• Eqn.16 shows the required speed of ψ to obtain certain desired values for the speeds ofφ and θ.

• Note that in the differential equations, the absolute values of the angles β _Ss ψ, φ.and θ exist alongside the rate of change of these angles. Therefore, the rate of change of any angle at any certain instant depends on the absolute values of the angles at that same instant.

2.3- Means of actuation o Simply the arc-shaped components in configuration II can be actuated using any of the suggested ideas used to actuate the arc-shaped components in configuration I. o As for rotating D2, it is simple enough and doesn't need any specific suggestions.

Simply, a motor can be installed directly on the hinge of H2 as shown in Fig.12b. o There are many other designs as shown in Fig.12c to satisfy the mirror rule, different configurations could be mixed and matched with different actuation methods. 3- Rotation matrix:

• A rotation matrix between Ll and L2* will be derived, since it is a useful tool in any robotic joint.

As shown in Fig.lθa

• In link Ll the origin is center point C, the axis Z ₁ is aligned with the axis of Ll, the axis X ₁ is aligned with the axis of D2-1, the axis Y ₁ is aligned with the axis of Dl-I, this will be referred to as Coordinate system 1.

• In link L2* the origin is point C, the axis Z _2* is aligned with the axis of L2*, the axis X _2* is aligned with the axis of D2-2, the axis Y _2* is aligned with the axis of D 1-2, this will be referred to as the Coordinate system 2*.

As shown in Fig.lOb

• Due to the difficulty of the direct derivation of the rotation matrix, a simple method will be used to derivate it, the real aim is to derivate the rotation matrix between two mechanical links where there is no twist angle between them, therefore, mechanism 1 (from the background art) will be recalled, review Fig.l, using its three degrees of freedom, not only two degrees of freedom like the new mechanism (which is the subject of this patent), in addition, mechanism 1 has a Locking delay drawback, but this will make no difference in the derivation process of the rotation matrix.

The axes X ₂ , Y ₂ , and Z ₂ are spatially fixed relative to KIa, the origin point is point C, axis Z ₂ is always aligned with Z ₁ , and this is referred to as coordinate system 2.

The axes X ₃ , Y ₃ , and Z ₃ are spatially fixed relative to KIb, the origin point is point C, axis Y ₃ is always aligned with Y ₂ , axis Z ₃ is always aliened with Z _2* and this is referred to as coordinate system 3.

In the initial position all axes X ₁ , X ₂ , X ₃ , and X _2* are aliened.

In the initial position all axes Y ₁ , Y ₂ , Y ₃ , and Y _2* are aliened.

In the initial position all axes Zi, Z ₂ , Z ₃ , and Z ₂ * are aliened.

Coordinate system 2 rotates an angle φ relative to coordinate system 1 about axis Z ₁ .

Coordinate system 3 rotates an angle θ relative to coordinate system 2 about axis Y ₂ .

Coordinate system 2* rotates an angle -φ relative to coordinate system 3 about axis Z ₃

(to eliminate the twist angle).

Matrix Tl converts coordinate system 2 to coordinate system 1,

Matrix T2 converts coordinate system 3 to coordinate system 2,

Matrix T3 converts coordinate system 2* to coordinate system 3,T3 = - —sφ cφ

Matrix T converts coordinate system 2* to coordinate system 1, and is a result of the multiplication of the tree matrices Tl, T2, and T3.

...Rotation matrix

The above rotation matrix considers only two degrees of freedom between Ll & L2*, the remaining degree of freedom (D3) was left out since it can be optional, and is very easy to consider. 4- Extra parts

• The mechanism can be completely covered for protection from ambient effects, like dirt or water.

• This cover could be made of one elastic piece, as shown in Fig.13 a, where one of the drawings shows a section view.

• The cover could be made of many rigid parts, as shown in Fig.13b, some drawings show a section view, each part has a similar part in the opposite half, each part is formed of a conical, and a spherical surface as shown in Fig.13 c, where all conical surfaces share their vertex which is also center point C, also all spherical surfaces share this same center point, hi the extreme positions all the conical surfaces (in each half) leans on each other, and the spherical surfaces work as a shield which completely covers the mechanism. Fig.13d shows how to assemble this cover parts in sequence.

Brief Description of Drawings

Fig.l (Shows mechanism 1 of the background art)

Ll and L2 are mechanical links desired to be ball-jointed, Kl is mechanism Kl of the background art, KIa and KIb are the parts that form Kl, Dl is the degree of freedom between Ll and KIa, as D2 is the degree of freedom between KIa and KIb, and D3 is the degree of freedom between K2b and L2. Fig.2 (Shows mechanism 2 of the background art)

Ll and L2 are mechanical links desired to be ball-jointed, K2 is mechanism K2 of the background art, K2a, K2b, and K2c are the parts that form K2, Dl is the degree of freedom between K2a and K2b, as D2 is the degree of freedom between K2b and K2c, and D3 is the degree of freedom between K2c and L2. Fig.3a (Sets an example for two mechanical links in an initial position)

Ll and L2 are mechanical links, where 1, 2, and 3 are numbers given to different longitudinal segments on each of the two mechanical links. Fig.3bl (Shows a zero deviation angle and a zero twist angle)

3, 4, and 5 are numbers given to different longitudinal segments on each of the two mechanical links, θ is the deviation angle and τ is the twist angle. Fig.3b2 (Shows a zero deviation angle and a twist angle)

2, 3, 4, and 5 are numbers given to different longitudinal segments on each of the two mechanical links, θ is the deviation angle and τ is the twist angle. Fig.3cl (Shows a deviation angle and a zero twist angle)

1, 7, and 8 are numbers given to different longitudinal segments on each of the two mechanical links, θ is the deviation angle and τ is the twist angle. Fig.3c2 (Shows a deviation angle and a twist angle)

1, 6, 7, and 8 are numbers given to different longitudinal segments on each of the two mechanical links, θ is the deviation angle and τ is the twist angle. Fig.3d (Shows a definition for both deviation and twist angle)

3, 4, 5, and 6 are numbers given to different longitudinal segments on each of the two mechanical links, θ is the deviation angle and τ is the twist angle, Plane P contains the axes of Ll and L2, plane Pl is the intersection of plane P with Ll, plane P2 is on L2 and it corresponds to Pl in the initial condition.

Fig.4a (Snapshot 1 in a sequence showing the locking delay drawback of mechanism Kl) Ll and L2 are mechanical links, D2 is a degree of freedom of mechanism Kl. 4, 5, and 6 are numbers given to different longitudinal segments on each of the two mechanical links. X, Y, and Z are points in space on a path for motion.

Fig.4b (Snapshot 2 in a sequence showing the locking delay drawback of mechanism Kl) Dl and D3 are degrees of freedom of mechanism Kl. 4, 5, and 6 are numbers given to different longitudinal segments on each of the two mechanical links. X, Y, and Z are points in space on a path for motion.

Fig.4c (Snapshot 3 in a sequence showing the locking delay drawback of mechanism Kl) D2 is a degree of freedom of mechanism Kl . 4, 5, 6, 7, and 8 are numbers given to different longitudinal segments on each of the two mechanical links. X, Y, and Z are points in space on a path for motion. Fig.4d (Snapshot 4 in a sequence showing the locking delay drawback of mechanism Kl)

4, 5, 6, 7, and 8 are numbers given to different longitudinal segments on each of the two mechanical links. X, Y ₅ and Z are points in space on a path for motion.

Fig.5a (Shows the undesired twist drawback of mechanism Kl and the role of D1& D2)

Ll and L2 are mechanical links, Dl, D2, D3 are the degrees of freedom of mechanism Kl ₅ as they are rotational degrees of freedom, each angle traveled by any of these degrees of freedom is written down in each example, deviation angle θ and twist angle τ are shown in each example as a result of the angles traveled by Dl & D2. Fig.5b (The undesired twist drawback of mechanism Kl and the role of Dl, D2, & D3)

Ll and L2 are mechanical links, Dl, D2, D3 are the degrees of freedom of mechanism Kl, as they are rotational degrees of freedom, each angle traveled by any of these degrees of freedom is written down in each example, deviation angle θ and twist angle τ are shown in each example as a result of the angles traveled by Dl, D2 & D3. Fig.όa (Shows the undesired twist drawback of mechanism K2 and the role of D1& D2)

Ll and L2 are mechanical links, Dl, D2, D3 are the degrees of freedom of mechanism K2, as they are rotational degrees of freedom, each angle traveled by any of these degrees of freedom is written down in each example, deviation angle θ and twist angle τ are shown in each example as a result of the angles traveled by Dl & D2. Fig.6b (The undesired twist drawback of mechanism K2 and the role of Dl, D2, & D3)

Ll and L2 are mechanical links, Dl, D2, D3 are the degrees of freedom of mechanism K2, as they are rotational degrees of freedom, each angle traveled by any of these degrees of freedom is written down hi each example, deviation angle θ and twist angle τ are shown hi each example as a result of the angles traveled by Dl, D2 & D3. Fig.7a (Shows the movement constrain drawback of mechanism K2)

Ll and L2 are mechanical links, K2a and K2c are parts of mechanism K2, Dl & D2 are degrees of freedom of K2, θ is the deviation angle, the arrows point out the collision zones. Fig.7b Shows the movement constrain drawback of mechanism K2)

Ll and L2 are mechanical links desired, Dl, D2, and D3 are the degrees of freedom of K2, θ is the deviation angle, and τ is the twist angle. Fig.8a (Shows Configuration I of the disclosure of the invention)

Ll and L2 are mechanical links desired to be ball-jointed, L2* is a transient link between Ll and L2, as L2* could be considered a part of Configuration I, Configuration I consists of: pyramid-shaped components Rl & R2, Arc-shaped components Al-I, A2-1, Al -2, & A2-2, where the degrees of freedom in Configuration I are Dl-I, D2-1, D 1-2, D2- 2, & D3 and the two (driven) hinges Hl & H2. Fig.8b (Shows alternate components for Configuration I)

Rl & R2 are the pyramid-shaped components, and their alternative components are Ol & 02 as they are the conical-shaped components, Al-I, A2-1, A 1-2, & A2-2 are the Arc- shaped components, the figure also shows the reason behind the names pyramid-shaped & conical-shaped. Fig.8c (Shows Configuration II of the disclosure of the invention)

Ll and L2 are mechanical links desired to be ball-jointed, L2* is a transient link between Ll and L2, as L2* could be considered a part of Configuration II, Configuration II consists of: pyramid-shaped components Rl & R2, Arc-shaped components Al-I & Al-2, where the degrees of freedom in Configuration II are Dl-I, D 1-2, & D3 and the hinge H2. Fig.8d (Shows the other form of Configuration II of the disclosure of the invention)

With its different kinds of arc-shaped components and the cross shaped part, and the alternate degree of freedom Dl*. Fig.9a (Shows the coordinate system between Ll and L2*)

Plane P contains the axes of Ll and L2*, Plane N is perpendicular to the axis of Ll ₅ the axis of L2* is projected on plane N, angles θ & φ defines the position of L2* relative to Ll ₅ θ is also the deviation angle, vector V and its opposite vector —V are both parallel to the axis ofD2-l. Fig.9b (Participates in the derivation of the mathematical relations) angle θ _s is measured in Plane P ₅ angle φ is measured in Plane N, the axis of Ll ₅ hinge Hl ₅ H2, and projection of axis of L2* are all illustrated, Plane M is called the mirror plane and it is formed by the axes of the hinges Hl & H2, C is the center point of the simulated ball-joint, vector U is perpendicular to vector V and its opposite vector -V, vector U also falls in plane N ₅ β _s is the angel traveled by the degree of freedom Dl-I ₅ and ζ _s is the angle traveled by D2-1, ψ _s is half the angle between the two arc-shaped components Al-I & Al -2, the distances λ, γ, and α are all used in the derivation. Fig.lOa (Participate in the derivation of the rotation matrix)

C is the center point, the axes X ₁ , Y ₁ , and Z ₁ form coordinate system 1 which is constrained to link Ll, and the axes X _2* , Y _2* , and Z _2* form coordinate system 2* which is constrained to L2*. Fig.lOb (Participate in the derivation of the rotation matrix)

C is the center point, the axes X ₁ , Y ₁ , and Z ₁ form coordinate system 1 which is constrained to link Ll, and the axes X _2* , Y _2* , and Z _2* form coordinate system 2* which is constrained to L2*, the axes X ₂ , Y ₂ , and Z ₂ form coordinate system 2 which is constrained to part KIa of mechanism Kl ₅ the axes X ₃ , Y ₃ , and Z ₃ form coordinate system 3 which is constrained to part KIb of mechanism Kl . Fig.lla (Shows an actuation method for the arc parts, that applies the mirror rule)

Pyramid-shaped parts Rl & R2 and arc- shaped parts Al-I and Al -2 are actuated using strings Sl and S2, as string S2 passes through the holes on the axis of hinge H2. Fig.llb (Shows how to gain symmetry using two hydraulic cylinders)

Where the hydraulic fluids used are Fl and F2. Fig.llc (Shows an actuation method for the arc parts, that applies the mirror rule)

Pyramid-shaped parts Rl & R2 are illustrated, as well as the arc- shaped parts Al-I and Al -2, strings S3 and S4 are used alongside a hydraulic fluid for actuation, where both strings are fixed on pins on hinge H2. Fig.l2a (Shows an actuation method for the arc parts, that applies the mirror rule)

Using only hydraulic fluids F3, F4, F5, and F6, and a valve with 4 different states, to rotate clockwise, counterclockwise, brake lock, or be passive. Fig.l2b (Shows an actuation method for the hinge of Configuration II)

Simply using a motor on the hinge of Configuration II. Fig.12c (Shows different designs for the configurations of the invention) Fig.l3a (Shows an elastic cover for the invention)

Fig.l3b (Shows a rigid cover made out of many solid pieces, to protect the invention) Fig.l3c (Shows the shape of the pieces of the rigid cover) Fig.l3d (Shows how to assemble the cover pieces of the rigid cover) Fig.14 (Visualize the different industrial application for the invention)

Industrial Applicability

This Mechanism could be used in any robotic or mechanical field, like the arms of industrial robots, manually operated or CNC machines, even in small applications like rotating a surveillance camera, or to rotate the joints of moving robots weather bipedal humanoid robots or others, also in military applications like rotating and aiming a tanks cannon or an airplane machinegun, also in communication applications like rotating a satellite dish, and in medical applications like artificial limbs.

Concisely this invention is a new mechanism, it is not exclusive for a specific application, so it is not possible to enumerate all its possible uses.

Fig.14 shows some visualization for different applications.