Cutter path smoothing system for robot machining

Abstract

The invention discloses a tool path fairing system for robot machining, which is used for improving path continuity in robot machining based on arc length parameterization path fairing calculation of a Clothoid curve and a B spline curve. In the path optimization process, a position path is transited at a corner by utilizing a pair of symmetrical Clothoid curves, and arc length parameterization is realized by combining the relationship between the arc length and a tangential angle; a direction path is constructed in an arc length-Euler angle space, a symmetrical B spline curve is adopted at each corner for transition, and finally arc length parameterization is completed; and the redundant angle is subjected to interpolation fitting through a B spline curve taking the arc length as a parameter. The arc length parameterization path fairing mode not only optimizes the continuity of the path, but also creates favorable conditions for subsequent feeding speed planning and interpolation point calculation, simplifies the calculation of the subsequent processing flow, and finally improves the processing efficiency and precision.

Description

Technical Field

[0001] This invention relates to the field of CNC machining, and in particular to a tool path smoothing system for robotic machining. Background Technology

[0002] Path smoothing refers to the technique of smoothing the tool path in CNC machining, with the aim of improving machining quality and efficiency.

[0003] With the rapid development of robotics technology, robots are gradually being applied in the processing of complex parts.

[0004] Typically, a given toolpath is represented by G01 code, which is a discrete straight-line path. At the endpoints of the line segment, the path has low continuity, only positional continuity. If the tool moves entirely along this path during machining, the abrupt change in direction at corners can lead to significant acceleration and impact during the movement.

[0005] The formation of "corners" at the junctions of adjacent line segments causes curvature discontinuities and sudden changes in feed rate (typically achieved by decelerating and then accelerating). This results in machine tool vibration and workpiece surface chatter marks due to frequent acceleration and deceleration, while also extending processing time. Therefore, it is necessary to optimize the straight-line path to improve its continuity within a given error range.

[0006] In robotic machining, the machining path includes position, orientation, and redundant angles;

[0007] The position is a three-dimensional point in Cartesian space, the direction is represented by two Euler angles, and the redundant angle is represented by one Euler angle.

[0008] Path smoothing requires not only smoothing the position, direction, and redundant angles separately, but also parameter synchronization so that the three can change synchronously during the movement, thereby improving the stability of the processing.

[0009] For example, Chinese patent CN202411929715.5 uses the arc length and curvature extrema of three pairs of Airthoid curves to obtain the smooth curve circumscribed by the tool path. Summary of the Invention

[0010] The main objective of this invention is to propose a toolpath smoothing system for robotic machining, which aims to improve the smoothness of the tool for discrete paths containing position, orientation, and redundant angle information based on Clothoid curves and B-spline curves.

[0011] To achieve the above objectives, this invention proposes a toolpath smoothing system for robotic machining, comprising:

[0012] Step 1: Determine the Clothoid curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path position;

[0013] Step 2: Determine the B-spline curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path direction;

[0014] Step 3: Use a quadratic B-spline curve to interpolate the path redundancy angle and generate an arc-length parameterized path redundancy angle.

[0015] The technical solution of this invention addresses the path location by using a pair of symmetrical Clothoid curves for local transition at each corner, and by utilizing the analytical relationship between the arc length and tangential angle of these curves to obtain the arc length parameterized path location.

[0016] For the path direction, two paths are constructed in the arc length-Euler angle space. A symmetrical B-spline curve is used at each corner to control the error in a coordinated manner, and finally the arc length parameterized path direction is obtained.

[0017] To address path redundancy angles, a B-spline curve is used to interpolate and fit the path in the arc-length-Euler angle space, thereby obtaining the arc-length parameterized path redundancy angle. This arc-length parameterized path smoothing not only improves the geometric properties of the path but also facilitates subsequent feed rate planning and interpolation point calculation, simplifies the calculation of subsequent machining processes, and ultimately improves machining efficiency and accuracy.

[0018] During path optimization, the position path utilizes a pair of symmetrical Clothoid curves for transition at corners, with arc length parameterized by combining the relationship between arc length and tangential angle. The directional path is constructed in arc length-Euler angle space, employing a symmetrical B-spline curve for transition at each corner, ultimately achieving arc length parameterization. Redundant angles are fitted using B-spline curve interpolation with arc length as the parameter. This arc length parameterized path smoothing method not only optimizes path continuity but also creates favorable conditions for subsequent feed rate planning and interpolation point calculation, simplifying the calculation of subsequent machining processes and ultimately improving machining efficiency and accuracy.

[0019] Meanwhile, the smooth toolpath design can serve as a typical model, which not only improves the linear transition of the path, but also simplifies the curve interpolation when parameters need to be modified. For example, by designing a predetermined model and using predetermined machining parameters as the basic values ​​and formulas, the predetermined curve interpolation can be modified when adjustments are needed, thereby achieving rapid toolpath calculation, which improves machining accuracy, reduces time costs, and increases machining efficiency. Attached Figure Description

[0020] Figure 1 This is the overall process of the present invention;

[0021] Figure 2a A path location diagram in Cartesian space;

[0022] Figure 2b This is an Euler angle diagram showing the path direction and redundant angles.

[0023] Figure 3 These are the smoothed results and magnified views of the path location;

[0024] Figure 4 This shows the smoothing results and a magnified view of the path direction;

[0025] Figure 5 It is the smoothing result of the path redundancy angle;

[0026] Figure 6 A table showing path location, direction, and redundant angles;

[0027] Figure 7 C i Table of maximum and final values ​​of transition length (s);

[0028] Figure 8 The cumulative arc length s i Value table;

[0029] Figure 9 For B α，j (u) and B β，j Table of temporary and final values ​​for the transition length of (u);

[0030] Figure 10 For the normalization parameter v i Value table;

[0031] Figure 11 For control point d i Value table;

[0032] Figure 12 A comparison chart of tangential velocity curves;

[0033] Figure 13 This represents the actual trajectory change at the end of the robotic arm. Detailed Implementation

[0034] The technical solutions in the embodiments of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0035] It should be noted that if the embodiments of the present invention involve directional indicators (such as up, down, left, right, front, back, top, bottom, inside, outside, vertical, horizontal, longitudinal, counterclockwise, clockwise, circumferential, radial, axial, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0036] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, features defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0037] like Figures 1 to 11 As shown, a toolpath smoothing system for robotic machining includes:

[0038] Step 1: Determine the Clothoid curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path position;

[0039] Suppose the given path location is represented by discrete three-dimensional points, denoted as {P0, P1, ..., P...}. n}(P i =[P x，i P y，i P z,i ]T),

[0040] Then the length of each line segment is

[0041] The angle between two adjacent line segments is

[0042] Therefore, there are n-1 corners along the path, located at point P. i (i = 1, ..., n-1) around.

[0043] At each corner, a pair of symmetrical Clothoid curves are used for local smoothing.

[0044] With straight line segments and Taking the corner as an example, the starting points of the two Clothoid curves are respectively on the straight line segment. and Above, denoted as B s，i and B e，i The specific expression is:

[0045]

[0046] Among them l i The transition length of the Clothoid curve is an unknown quantity that needs to be determined.

[0047]

[0048] Let be the tangential unit vector of the Clothoid curve.

[0049] Norm() is the normalization function. The expression for the Clothoid curve is:

[0050]

[0051] in, is the normal unit vector of the Clothoid curve.

[0052] C(θ) and S(θ) are Fresnel integrals, expressed as follows:

[0053]

[0054] a i The scaling factor for the Clothoid curve is l. i The following relationship exists:

[0055]

[0056] Once l is determined i The value of ε allows us to determine the expression for the transition curve. Let the upper limit of the given position profile error be ε. p，max Based on the deviation relationship between the Clothoid curve and the original straight line segment, l can be determined. i upper limit l max，i as follows:

[0057]

[0058] Under the constraint of contour error, l i tentatively set as l max，i .

[0059] Subsequently, overlap avoidance is considered for l i Adjustments will be made. Specifically, P i The condition for the transition curves to overlap on the front side is:

[0060]

[0061] The condition for overlap at the rear is:

[0062]

[0063] If (6) is true and (7) is false, then

[0064]

[0065] If (7) is true and (6) is false, then

[0066]

[0067] If both (6) and (7) are true, then

[0068]

[0069] If neither (6) nor (7) is true, then l i =l max，i This ensures that the error of the transition curve at each corner is less than a given threshold, and that adjacent transition curves do not overlap.

[0070] Based on the relationship between the arc length s of the Clothoid curve and the tangential angle θ, its parameterized expression for the arc length can be obtained:

[0071]

[0072] The parameterized expression for the arc length of the transition curve at the corner is:

[0073]

[0074] Let the midpoint of each transition curve be... Its arc length along the entire smooth path is s i (i = 1, 2, ..., n-1),

[0075] Points P0 and P n The arc lengths along the entire smooth path are s0 and s. n The calculation formula is as follows:

[0076]

[0077] Among them, |P0B s，1 | represents a line segment The length of is similar for the others.

[0078] In summary, the parameterized expression for the arc length of the entire smooth path is:

[0079]

[0080]

[0081] Step 2: Determine the B-spline curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path direction;

[0082] Let the given path direction be represented by Euler angles, denoted as {α0, α1, ..., α...}. n} and {β0, β, ..., β n}

[0083] Then there exist two paths CL in arc length-Euler angle space. α ={(s i α i )} and CL β ={(s i ,β i )}.

[0084] At each corner, a symmetrical B-spline curve is used for local smoothing.

[0085] With CL α and CL β Taking the i-th corner as an example, the two B-spline curves used for local smoothing are as follows:

[0086]

[0087] in, h i This represents the transition length of these two transition curves.

[0088] N j，3 (u) is a basis function defined on the node vector U = [0 0 0 0 0.5 1 1 1 1], R α，j and R β，j These are the control points. To ensure the second-order continuity of the transition curve at its endpoints, the control points are:

[0089]

[0090] in,

[0091] Similarly, as long as h is determined i B α，j (u) and B β，j The expression for (u) is then completely determined.

[0092] Based on the constraint of no overlap, the transition length h i It should satisfy:

[0093]

[0094] Based on the deviation relationship between the B-spline curve and the original straight line segment, the transition error can be obtained as follows:

[0095]

[0096] in,

[0097] Let the upper limit of the given position contour error be ε. E，max Under the constraints of contour error and non-overlap, the transition length h i tentatively set as

[0098]

[0099] Subsequently, adjustments are made to the transition curve, which can be further expanded. Specifically, for the i-th corner, if the following conditions are met:

[0100]

[0101] Then h i It can be further updated to:

[0102]

[0103] In summary, the parameterized expression for the arc length along the entire smooth path is:

[0104]

[0105]

[0106] Step 3: Use a quadratic B-spline curve to interpolate the path redundancy angle and generate an arc-length parameterized path redundancy angle.

[0107] Let the given path redundancy angles be represented by Euler angles, denoted as {γ0, γ1, ..., γ...} n If}, then there exists a path CL in arc-length-Euler angle space. γ ={(s i γ i These discrete points are interpolated using a quadratic B-spline curve, thereby improving path continuity.

[0108] First, for point (s) i γ i ) Perform arc length parameterization, that is:

[0109]

[0110] Based on this, the initial node vector V0 = [t0 t1……t] is determined. n+3 t n+4 ]as follows:

[0111]

[0112] Further, insert a new node between two adjacent and distinct nodes.

[0113] The final node vector V = [μ0 μ1……μ] is obtained. 2n+3 μ 2n+4 ]as follows:

[0114]

[0115] The expression for the quadratic B-spline curve D(v) is:

[0116]

[0117] Among them, b j，2 (v) is a basis function defined on the node vector V. d j For control points.

[0118] The geometric smoothness of D(v) is defined as:

[0119]

[0120] Where, D = [d0 d1…d 2n d 2n+1 ] T Let E be the column vector consisting of the control points, and let E be the inertia matrix.

[0121] Each element is respectively

[0122] For each point (v) i ,γ i ),

[0123] The interpolation condition is [b 0,2 (v i )b 1,2 (v i ...b 2n,2 (v i )b 2n+1，2 (v i )]D=γ i (i = 0, 1, ... n) (28);

[0124] Furthermore, the first and second derivatives of the starting and ending points are 0, that is:

[0125]

[0126] In summary, the optimization problem is as follows:

[0127] in,

[0128] The control points {d0 d1…d} were obtained by solving this problem using optimization tools. 2n d 2n+1 The specific value of}.

[0129] In summary, the parameterized expression for the arc length of the redundant angle along the entire smooth path is:

[0130]

[0131] The overall process of this invention is as follows: Figure 1 As shown,

[0132] The steps of the present invention will be implemented below using a path as shown in Figure 2 as an example.

[0133] Figure 2(a) shows the path location in Cartesian space. In Figure 2(b), the vertical axis represents the Euler angles corresponding to the path direction and redundant angles, while the horizontal axis represents the index of the path point.

[0134] The location, direction, and redundancy angle values ​​of the path are as follows: Figure 6 As shown:

[0135] like Figure 6 As shown: path location, direction, and redundancy angle.

[0136] The position contour error ε given in step one p，max =0.05mm, based on position P i The length L of the straight line segment is calculated. i The angle 2θ between two adjacent line segments i .

[0137] Substituting the relevant data into formula (5), we obtain the upper limit value l of the transition length of the transition curve. max，i ;

[0138] like Figure 7 As shown.

[0139] If the upper limit value is used directly, overlap may occur. By judging whether formulas (6) and (7) are valid, adjustments are made according to the corresponding formulas (8-10) to obtain l. i The final value is shown in Figure 2. Then, s is calculated according to formula (13). i ,like Figure 3 As shown. Finally, based on formulas (11-12) and (14), the parameterized expression P(s) of the arc length of the entire smooth path position is obtained, as follows. Figure 3 As shown by the dashed line, the solid line represents the original straight line path.

[0140] Figure 7 As shown: C i The maximum and final determined value of the transition length of (s).

[0141] Figure 8 As shown: Cumulative arc length s i The value of .

[0142] The upper limit of position contour error given in step 2 The temporary value of the transition length can be determined according to formula (19). Then, a judgment is made according to formula (20). If the condition is met, the value of the transition length h is updated according to formula (21) to obtain the value of the transition length h. i like Figure 4 As shown.

[0143] Finally, based on formulas (15-16) and (22), the parameterized expression for the arc length O(s) along the entire smooth path direction is obtained, as follows: Figure 9 The dashed lines in the diagram represent the original straight line path, while the solid lines represent the original straight line path.

[0144] Figure 4 B α,j (u) and B β,j The temporary and final values ​​of the transition length of (u).

[0145] In step 3, the arc length is first parameterized according to formula (23) to obtain the normalized parameter v corresponding to each redundant angle. i like Figure 5 As shown.

[0146] Then, according to formula (24-25),

[0147] Obtain the node vector:

[0148] V=[0,0,0,0.0502,0.1004,0.1506,0.2008,0.2509,0.3011,0.3512,0.4013,0. 4514,0.5015,0.5516,0.6017,0.6517,0.7017,0.7680,0.8344,0.9172,1,1,1].

[0149] Construct an optimization problem as shown in formula (30), and solve it to obtain the control point d. i Values ​​such as Figure 6 As shown.

[0150] Finally, based on formulas (26) and (31), the parameterized expression γ(s) for the arc length of the redundancy angle of the entire smooth path is obtained.

[0151] like Figure 10As shown by the dashed line, the solid line represents the original straight line path.

[0152] like Figure 10 As shown: Normalized parameter v i The value of .

[0153] like Figure 11 As shown: Control point d i The value of .

[0154] Applying the toolpath smoothing system of this application to the PUMA560 and UR5 robotic arms, the machining efficiency can be improved by using this path smoothing method under the same geometric error constraints and kinematic upper limit constraints (which can be understood as the same machining accuracy requirements) and the same speed planning method.

[0155] Specifically, the PUMA560 is a series of industrial robots gradually launched by the American company Unimation (later acquired by the Swiss company ABB) in 1978, originally designed for automated assembly tasks in the manufacturing industry.

[0156] Structural features and degrees of freedom: a flexible arm with six degrees of freedom, all joints being rotational joints.

[0157] Joint axis relationships: z1 and z2 intersect, z2 and z3 are parallel, and z4, z5, and z6 intersect.

[0158] Additionally, the UR5 is a six-axis collaborative robot developed by the Danish company Universal Robots.

[0159] Structural features and degrees of freedom: Six degrees of freedom; the joint structure design gives it high flexibility and operability.

[0160] Joint axis relationship: The axes of the 2nd, 3rd, and 4th joints are parallel, satisfying Pieper's criterion.

[0161] Specifically, the system was compared with existing methods on the PUMA560 robotic arm.

[0162] The obtained tangential velocity curve is as follows Figure 12 As shown, the red curve is the speed curve on the smooth path obtained by this method, while the blue and black curves are the speed curves obtained by two existing methods. It can be seen that this method can significantly improve processing efficiency.

[0163] exist Figure 12 In the test,

[0164] The timeline on the X-axis clearly shows a 20% increase in processing efficiency;

[0165] The cutting speed on the Y-axis can be obtained, and the speed can be faster at the "corner" position, thus avoiding problems such as tool marks and vibration caused by speed reduction. It can be understood that if the cutting speed and tool speed are constant during the cutting process, the precision of the product will be higher. However, in the current processing of complex products, "corners" are inevitable (and there are many types of corners). Therefore, this system can effectively improve processing efficiency.

[0166] Experiments were conducted on the UR5 robotic arm, and the changes in the robotic arm were as follows: Figure 13 As shown, the actual trajectory of the robotic arm's end effector is as follows: Figure 13 As shown by the red dot.

[0167] It can be clearly seen that the speed curve transition of its path wear is more linear, which effectively reduces the vibration of tool speed switching, reduces tool marks and other problems, and improves the stability and accuracy of precision machining.

[0168] The above description is merely a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A toolpath smoothing system for robotic machining, characterized in that, include, Step 1: Determine the Clothoid curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path position; Step 2: Determine the B-spline curve under the constraints of contour error and non-overlap, and generate the arc length parameterized path direction; Step 3: Use a quadratic B-spline curve to interpolate the path redundancy angle and generate an arc-length parameterized path redundancy angle.

2. The toolpath smoothing system for robot machining as described in claim 1, characterized in that: In step one: Suppose the given path location is represented by discrete three-dimensional points, denoted as {P0, P1, ..., P2}. n }(P i =[P x，i P y，i , P z，i ] T ), Then the length of each line segment is The angle between two adjacent line segments is There are n-1 corners along the path, located at point P. i The position is near (i = 1, ..., n-1); at each corner, a pair of symmetrical Clothoid curves are used for local smoothing.

3. The toolpath smoothing system for robot machining as described in claim 2, characterized in that: With straight line segments and Using the corner formed as the base point, the starting points of the two Clothoid curves are respectively on the straight line segment. and Above, denoted as B s，i and B e，i , The specific expression is: Among them, l i The transition length of the Clothoid curve is an unknown quantity that needs to be determined. Let be the tangential unit vector of the Clothoid curve. Norm() is the normalization function; the expression for the Clothoid curve is: in, is the normal unit vector of the Clothoid curve.

4. The toolpath smoothing system for robot machining as described in claim 3, characterized in that: C(θ) and S(θ) are Fresnel integrals, expressed as follows: a i The scaling factor for the Clothoid curve is l. i The following relationship exists: Determine i The value of is fixed, and the expression for the transition curve is a definite value.

5. The toolpath smoothing system for robot machining as described in claim 4, characterized in that: When the upper limit of the given position contour error is ε p，max , Based on the deviation relationship between the Clothoid curve and the original straight line segment, l can be determined. i upper limit l max，i as follows: Under the constraint of contour error, l i tentatively set as l max，i ; The P i The condition for the transition curves to overlap on the front side is: The condition for overlap at the rear is: If (6) is true and (7) is false, then If (7) is true and (6) is false, then If both (6) and (7) are true, then If neither (6) nor (7) is true, then l i =l max，i , When the error of the transition curve at each corner is less than a given threshold, and adjacent transition curves do not overlap; Based on the relationship between the arc length s of the Clothoid curve and the tangential angle θ, the parameterized expression for the arc length is as follows: The parameterized expression for the arc length of the transition curve at the corner is: When the midpoint of each transition curve is The arc length on the smooth path is s. i (i = 1, 2, ..., n-1), Points P0 and P n The arc lengths along the entire smooth path are s0 and s. n The calculation formula is as follows: Among them, |P0B s,1 | represents a line segment Length; The parameterized expression for the arc length of the entire smooth path is:

6. The toolpath smoothing system for robot machining as described in claim 5, characterized in that: In step two, the path direction is represented by Euler angles, which are {α0, α1, ..., α...} n } and {β0,β,...,β n }; Then there exist two paths CL in arc length-Euler angle space. α ={(s i ,α i )} and CL β ={(s i ,β i )}; At each corner, a symmetrical B-spline curve is used for local smoothing; With CL α and CL β Taking the i-th corner as an example, the two B-spline curves used for local smoothing are as follows: in, h i This represents the transition length of these two transition curves.

7. The toolpath smoothing system for robot machining as described in claim 6, characterized in that: N j，3 (u) is a basis function defined on the node vector U = [0 0 0 0 0.5 1 1 1 1], R α，j and R β，j For control points; When the transition curve exhibits second-order continuity at its endpoints, the control points are: in, When h is determined i hour, The B α，j (u) and B β，j The expression for (u) is then determined.

8. The toolpath smoothing system for robot machining as described in claim 7, characterized in that: Based on the constraint of no overlap, the transition length h i It should satisfy: Based on the deviation relationship between the B-spline curve and the original straight line segment, the transition error can be obtained as follows: in, When the upper limit of the given position contour error is ε E，max Under the constraints of contour error and non-overlap, the transition length h i tentatively set as Subsequently, adjustments were made to the transition curve, which could be further expanded. For the i-th corner, if the following conditions are met: Then h i It can be further updated to: The parameterized expression for the arc length along the smooth path direction is:

9. The toolpath smoothing system for robot machining as described in claim 8, characterized in that: Step three The path redundancy angle is represented by Euler angles, namely {γ0, γ1, ..., γ...} n If}, then there exists a path Cl in arc-Euler angle space. γ ={(s i γ i )}; These discrete points are interpolated using a single quadratic B-spline curve. Point (s) i γ i ) Perform arc length parameterization, that is: Based on this, the initial node vector V0 = [t0 t1 ...... t] is determined. n+3 t n+4 ]as follows: Insert a new node between two adjacent and distinct nodes. The final node vector V = [μ0 μ1 ...... μ] is obtained. 2n+3 μ 2n+4 ]as follows: The expression for the quadratic B-spline curve D(v) is: Among them, b j，2 (v) is a basis function defined on the node vector V. d j For control points; The geometric smoothness of D(v) is defined as: in, D = [d0, d1...d2] 2n d 2n+1 ] T Let E be the column vector consisting of the control points, and let E be the inertia matrix. Each element is respectively Each point (v) i γ i The interpolation condition is: [b 0.2 (v i )b 1，2 (v i )...b 2n，2 (v i )b 2n+1，2 (v i )]D=γ i (i=0,1,..n) (28); When the first and second derivatives of the starting and ending points are 0, that is: The optimized formula is as follows: in, The control points {d0d1...d} are solved using optimization formulas. 2n d 2n+1 The specific value of}; The parameterized expression for the arc length of the redundancy angle of the entire smooth path is:

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